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%!PS-Adobe-2.0 %%Creator: dvips(k) 5.991 Copyright 2011 Radical Eye Software %%Title: arXiv:1301.4970v2 [math.AG] 5 May 2014 %%CreationDate: Sat Nov 18 18:27:43 2023 %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: EUFM10 EUFM7 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z diagonalv4.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2023.11.18:1827 %%BeginProcSet: texc.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat {BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B /M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ 0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Add) cvn H.B /ANN pdfmark end 2608 2090 a Fv(]\).)77 b(T)-8 b(aking)42 b(the)g(cones)h(of)g(the)75 2198 y(F)-8 b(ourier{Muk)j(ai)35 b(transforms)g(indeed)f(mak)m(es)i(sense,)h(since)e(all)f(the)i(o)s (ccurring)e(maps)h(are)h(induced)d(b)m(y)75 2306 y(maps)e(b)s(et)m(w)m (een)h(the)g(in)m(tegral)g(k)m(ernels)f(\(see)h([)1670 2306 y SDict begin H.S end 1670 2306 a Fv(AL12)1886 2243 y SDict begin H.R end 1886 2243 a 1886 2306 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.AL) cvn H.B /ANN pdfmark end 1886 2306 a Fv(]\).)45 b(W)-8 b(e)33 b(set)f(k)m(er)16 b Ft(R)28 b Fv(:=)f Fr(f)p Ft(B)33 b Fr(2)26 b(B)k(j)e Ft(R)q(B)j Fv(=)c(0)p Fr(g)p Fv(.)46 b(By)32 b(the)75 2416 y(adjoin)m(t)e(prop)s(ert)m(y)g(it)g(equals)f(the)i(righ)m (t-orthogonal)g(complemen)m(t)f(\(im)15 b Ft(F)e Fv(\))2718 2383 y Fo(?)2777 2416 y Fv(.)75 2578 y SDict begin H.S end 75 2578 a 75 2578 a SDict begin 13 H.A end 75 2578 a 75 2578 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.1) cvn /DEST pdfmark end 75 2578 a FF(Prop)s(osition)36 b(2.1)g Fv(\([)869 2578 y SDict begin H.S end 869 2578 a Fv(Add11)1129 2515 y SDict begin H.R end 1129 2515 a 1129 2578 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Add) cvn H.B /ANN pdfmark end 1129 2578 a Fv(,)31 b(section)f(3]\))p FF(.)43 b Fl(L)-5 b(et)33 b Ft(F)23 b Fv(:)31 b Fr(A)25 b(!)g(B)35 b Fl(b)-5 b(e)32 b(a)h Fs(P)2529 2545 y Fq(n)2576 2578 y Fl(-functor.)75 2619 y SDict begin H.S end 75 2619 a 75 2619 a SDict begin 13 H.A end 75 2619 a 75 2619 a SDict begin [/View [/XYZ H.V]/Dest (Item.4) cvn /DEST pdfmark end 75 2619 a 273 2705 a Fv(\(i\))42 b Fl(We)32 b(have)h Ft(P)843 2719 y Fq(F)902 2705 y Fv(\()p Ft(B)5 b Fv(\))25 b(=)g Ft(B)37 b Fl(for)c Ft(B)d Fr(2)25 b Fv(k)m(er)16 b Ft(R)q Fl(.)75 2727 y SDict begin H.S end 75 2727 a 75 2727 a SDict begin 13 H.A end 75 2727 a 75 2727 a SDict begin [/View [/XYZ H.V]/Dest (Item.5) cvn /DEST pdfmark end 75 2727 a 248 2813 a Fv(\(ii\))41 b Ft(P)469 2827 y Fq(F)548 2813 y Fr(\016)20 b Ft(F)39 b Fr(')25 b Ft(F)33 b Fr(\016)21 b Ft(H)1046 2780 y Fq(n)p Fp(+1)1183 2813 y Fv([2])p Fl(.)75 2835 y SDict begin H.S end 75 2835 a 75 2835 a SDict begin 13 H.A end 75 2835 a 75 2835 a SDict begin [/View [/XYZ H.V]/Dest (Item.6) cvn /DEST pdfmark end 75 2835 a 223 2921 a Fv(\(iii\))40 b Fl(The)32 b(obje)-5 b(cts)33 b(in)g Fv(im)14 b Ft(F)33 b Fr([)20 b Fv(k)m(er)c Ft(R)33 b Fl(form)h(a)f(sp)-5 b(anning)34 b(class)f(of)g Fr(B)s Fl(.)75 2943 y SDict begin H.S end 75 2943 a 75 2943 a SDict begin 13 H.A end 75 2943 a 75 2943 a SDict begin [/View [/XYZ H.V]/Dest (Item.7) cvn /DEST pdfmark end 75 2943 a 225 3028 a Fv(\(iv\))42 b Ft(P)469 3042 y Fq(F)560 3028 y Fl(is)33 b(an)g(auto)-5 b(e)g(quivalenc)g(e.)75 3105 y SDict begin H.S end 75 3105 a 75 3105 a SDict begin 13 H.A end 75 3105 a 75 3105 a SDict begin [/View [/XYZ H.V]/Dest (Item.8) cvn /DEST pdfmark end 75 3105 a 91 x FF(Example)34 b(2.2.)240 b Fv(\(i\))41 b(Let)31 b Fr(B)d Fv(=)d(D)1462 3160 y Fq(b)1496 3196 y Fv(\()p Ft(X)7 b Fv(\))32 b(for)e(a)g(smo)s (oth)g(pro)5 b(jectiv)m(e)31 b(v)-5 b(ariet)m(y)30 b Ft(X)7 b Fv(.)41 b(A)30 b Fs(P)3254 3163 y Fq(n)3301 3196 y Fl(-obje)-5 b(ct)39 b Fv(\(see)411 3303 y([)436 3303 y SDict begin H.S end 436 3303 a Fv(HT06)661 3241 y SDict begin H.R end 661 3241 a 661 3303 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.HT) cvn H.B /ANN pdfmark end 661 3303 a Fv(]\))32 b(is)f(an)h(ob)5 b(ject)33 b Ft(E)g Fr(2)28 b(B)34 b Fv(suc)m(h)e(that)g Ft(E)27 b Fr(\012)21 b Ft(!)2182 3317 y Fq(X)2277 3303 y Fr(')27 b Ft(E)38 b Fv(and)31 b(Ext)2803 3268 y Fo(\003)2843 3303 y Fv(\()p Ft(E)5 b(;)15 b(E)5 b Fv(\))3126 3278 y Fr(\030)3126 3308 y Fv(=)3225 3303 y(H)3293 3268 y Fo(\003)3332 3303 y Fv(\()p Fs(P)3438 3270 y Fq(n)3486 3303 y Ft(;)15 b Fs(C)p Fv(\))33 b(as)411 3411 y Fs(C)p Fv(-algebras)24 b(\(the)h(ring)f(structure)g(on)g(the)h(left-hand)e(side)h(is)f(the)i (Y)-8 b(oneda)25 b(pro)s(duct)e(and)h(on)h(the)411 3519 y(righ)m(t-hand)h(side)g(the)h(cup)f(pro)s(duct\).)39 b(A)27 b Fs(P)1932 3486 y Fq(n)1979 3519 y Fv(-ob)5 b(ject)29 b(can)e(b)s(e)g(iden)m(ti\014ed)e(with)h(the)h Fs(P)3387 3486 y Fq(n)3434 3519 y Fv(-functor)1349 3672 y Ft(F)c Fv(:)46 b(D)1570 3634 y Fq(b)1605 3672 y Fv(\(pt\))25 b Fr(!)h(B)93 b Ft(;)106 b Fs(C)26 b Fr(7!)f Ft(E)411 3818 y Fv(with)k Ft(H)k Fv(=)26 b([)p Fr(\000)p Fv(2].)42 b(Note)32 b(that)g(the)f(righ)m(t)f(adjoin)m(t)h(is)f(indeed)f(giv)m (en)i(b)m(y)f Ft(R)d Fv(=)f(Ext)3243 3782 y Fo(\003)3282 3818 y Fv(\()p Ft(E)5 b(;)p 3429 3833 61 4 v 77 w Fv(\).)42 b(The)411 3926 y Fs(P)482 3893 y Fq(n)529 3926 y Fv(-t)m(wist)35 b(asso)s(ciated)g(to)g(the)g(functor)f Ft(F)48 b Fv(is)34 b(the)g(same)h(as)g(the)g Fs(P)2764 3893 y Fq(n)2811 3926 y Fv(-t)m(wist)g(asso)s(ciated)g(to)h(the)411 4034 y(ob)5 b(ject)31 b Ft(E)k Fv(as)c(de\014ned)e(in)g([)1344 4034 y SDict begin H.S end 1344 4034 a Fv(HT06)1569 3972 y SDict begin H.R end 1569 3972 a 1569 4034 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.HT) cvn H.B /ANN pdfmark end 1569 4034 a Fv(].)75 4057 y SDict begin H.S end 75 4057 a 75 4057 a SDict begin 13 H.A end 75 4057 a 75 4057 a SDict begin [/View [/XYZ H.V]/Dest (Item.9) cvn /DEST pdfmark end 75 4057 a 248 4142 a Fv(\(ii\))41 b(A)30 b Fs(P)580 4109 y Fp(1)620 4142 y Fv(-functor)g(is)f(the)i(same) g(as)f(a)h Fl(spheric)-5 b(al)35 b(functor)40 b Fv(\(see)32 b([)2539 4142 y SDict begin H.S end 2539 4142 a Fv(Ann07)2799 4080 y SDict begin H.R end 2799 4080 a 2799 4142 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Ann) cvn H.B /ANN pdfmark end 2799 4142 a Fv(]\))f(where)f(the)g(unit)1620 4305 y(id)1743 4248 y Fq(\021)1721 4305 y Fr(\000)-69 b(!)25 b Ft(R)q(F)38 b Fr(!)26 b Ft(H)411 4451 y Fv(splits.)38 b(In)30 b(this)f(case)j(there)e(is)g(also)g(the)h Fl(spheric)-5 b(al)34 b(twist)40 b Fv(giv)m(en)30 b(b)m(y)1426 4619 y Ft(T)1479 4633 y Fq(F)1563 4619 y Fv(:=)c(cone)1876 4518 y Fk(\020)1931 4619 y Ft(F)13 b(R)2120 4568 y Fq(")2097 4619 y Fr(\000)-71 b(!)25 b Fv(id)2289 4518 y Fk(\021)2373 4619 y Ft(:)411 4800 y Fv(It)30 b(is)f(again)i(an)f(auto)s(equiv)-5 b(alence)30 b(with)f Ft(T)1901 4767 y Fp(2)1888 4827 y Fq(F)1972 4800 y Fv(=)c Ft(P)2126 4814 y Fq(F)2216 4800 y Fv(\(see)31 b([)2423 4800 y SDict begin H.S end 2423 4800 a Fv(Add11)2683 4737 y SDict begin H.R end 2683 4737 a 2683 4800 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Add) cvn H.B /ANN pdfmark end 2683 4800 a Fv(,)g(p.)40 b(33]\).)75 4881 y SDict begin H.S end 75 4881 a 75 4881 a SDict begin 13 H.A end 75 4881 a 75 4881 a SDict begin [/View [/XYZ H.V]/Dest (Item.10) cvn /DEST pdfmark end 75 4881 a 81 x FF(Lemma)33 b(2.3.)240 b Fv(\(i\))41 b Fl(L)-5 b(et)40 b Fv(\011)c Fr(2)h Fv(Aut\()p Fr(A)p Fv(\))i Fl(such)h(that)g Fv(\011)24 b Fr(\016)i Ft(H)44 b Fr(')36 b Ft(H)c Fr(\016)26 b Fv(\011)p Fl(.)60 b(Then)40 b Ft(F)e Fr(\016)25 b Fv(\011)39 b Fl(is)g(again)h(a)411 5070 y Fs(P)482 5037 y Fq(n)529 5070 y Fl(-functor)32 b(with)i(the)f(pr)-5 b(op)g(erty)1670 5216 y Ft(P)1728 5230 y Fq(F)10 b Fo(\016)p Fp(\011)1902 5216 y Fr(')25 b Ft(P)2056 5230 y Fq(F)2130 5216 y Ft(:)p eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 67 -8 a SDict begin H.S end 67 -8 a 67 -8 a SDict begin H.R end 67 -8 a 67 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.4) cvn /DEST pdfmark end 67 -8 a 75 66 a Fp(4)1508 b(ANDREAS)24 b(KR)n(UG)75 183 y SDict begin H.S end 75 183 a 75 183 a SDict begin 13 H.A end 75 183 a 75 183 a SDict begin [/View [/XYZ H.V]/Dest (Item.11) cvn /DEST pdfmark end 75 183 a 248 266 a Fv(\(ii\))41 b Fl(L)-5 b(et)38 b Fv(\010)10 b(:)32 b Fr(B)38 b(!)c(C)44 b Fl(b)-5 b(e)37 b(an)i(e)-5 b(quivalenc)g(e)38 b(of)g(triangulate)-5 b(d)40 b(c)-5 b(ate)g(gories.)59 b(Then)39 b Fv(\010)23 b Fr(\016)i Ft(F)51 b Fl(is)38 b(again)h(a)411 374 y Fs(P)482 341 y Fq(n)529 374 y Fl(-functor)32 b(with)i(the)f(pr)-5 b(op)g(erty)35 b(that)1520 537 y Ft(P)1578 551 y Fp(\010)p Fo(\016)p Fq(F)1744 537 y Fr(\016)20 b Fv(\010)25 b Fr(')g Fv(\010)20 b Fr(\016)h Ft(P)2206 551 y Fq(F)2280 537 y Ft(:)75 709 y Fl(Pr)-5 b(o)g(of.)43 b Fv(The)27 b(pro)s(of)h(is)f (analogous)h(to)g(the)h(pro)s(of)e(of)h(the)g(corresp)s(onding)e (statemen)m(t)k(for)d(spherical)g(func-)75 817 y(tors)k([)283 817 y SDict begin H.S end 283 817 a Fv(Ann07)543 755 y SDict begin H.R end 543 755 a 543 817 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Ann) cvn H.B /ANN pdfmark end 543 817 a Fv(,)f(Prop)s(osition)f(2].)2496 b Fj(\003)75 989 y SDict begin H.S end 75 989 a 75 989 a SDict begin 13 H.A end 75 989 a 75 989 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.4) cvn /DEST pdfmark end 75 989 a FF(Corollary)58 b(2.4.)53 b Fl(L)-5 b(et)50 b Ft(E)1019 1003 y Fp(1)1058 989 y Ft(;)15 b(:)g(:)g(:)i(;)e(E)1327 1003 y Fq(n)1431 989 y Fr(2)57 b(B)52 b Fl(b)-5 b(e)50 b(a)g(c)-5 b(ol)5 b(le)-5 b(ction)51 b(of)g(p)-5 b(airwise)51 b(ortho)-5 b(gonal)53 b(\(that)f(me)-5 b(ans)75 1097 y Fv(Hom)264 1062 y Fo(\003)304 1097 y Fv(\()p Ft(E)406 1111 y Fq(i)434 1097 y Ft(;)15 b(E)541 1111 y Fq(j)579 1097 y Fv(\))38 b(=)g(0)h(=)f(Hom)1143 1062 y Fo(\003)1183 1097 y Fv(\()p Ft(E)1285 1111 y Fq(j)1321 1097 y Ft(;)15 b(E)1428 1111 y Fq(i)1457 1097 y Fv(\))40 b Fl(for)h Ft(i)d Fr(6)p Fv(=)g Ft(j)5 b Fl(\))41 b Fs(P)2054 1064 y Fq(n)2101 1097 y Fl(-obje)-5 b(cts)40 b(with)g(asso)-5 b(ciate)g(d)42 b(twists)f Ft(p)3381 1111 y Fq(i)3447 1097 y Fv(:=)e Ft(P)3640 1111 y Fq(E)3692 1121 y Fz(i)3722 1097 y Fl(.)75 1205 y(Then)952 1331 y Fs(Z)1017 1294 y Fq(n)1089 1331 y Fr(!)26 b Fv(Aut\()p Fr(A)p Fv(\))93 b Ft(;)108 b Fv(\()p Ft(\025)1817 1345 y Fp(1)1857 1331 y Ft(;)15 b(:)g(:)g(:)i(;)e(\025) 2112 1345 y Fq(n)2159 1331 y Fv(\))26 b Fr(7!)f Ft(p)2382 1291 y Fq(\025)2423 1300 y Fx(1)2382 1357 y Fp(1)2482 1331 y Fr(\016)20 b(\001)15 b(\001)g(\001)22 b(\016)e Ft(p)2785 1294 y Fq(\025)2826 1302 y Fz(n)2785 1354 y Fq(n)75 1476 y Fl(de\014nes)33 b(a)g(gr)-5 b(oup)34 b(isomorphism)h Fs(Z)1306 1443 y Fq(n)1378 1451 y Fr(\030)1378 1480 y Fv(=)1474 1476 y Fr(h)p Ft(p)1555 1490 y Fp(1)1595 1476 y Ft(;)15 b(:)g(:)g(:)h(;)f(p)1842 1490 y Fq(n)1890 1476 y Fr(i)25 b(\032)g Fv(Aut\()p Fr(B)s Fv(\))p Fl(.)75 1648 y(Pr)-5 b(o)g(of.)43 b Fv(By)24 b(part)f(\(ii\))g(of)h(the)g (previous)e(lemma)h(the)h Ft(p)1911 1662 y Fq(i)1963 1648 y Fv(comm)m(ute)g(whic)m(h)f(means)g(that)i(the)e(map)h(is)e (indeed)75 1766 y(a)29 b(group)f(homomorphism)e(on)m(to)k(the)f (subgroup)e(generated)i(b)m(y)g(the)g Ft(p)2534 1780 y Fq(i)2561 1766 y Fv(.)40 b(Let)30 b Ft(g)e Fv(=)d Ft(p)3001 1725 y Fq(\025)3042 1734 y Fx(1)3001 1792 y Fp(1)3098 1766 y Fr(\016)17 b(\001)e(\001)g(\001)i(\016)g Ft(p)3390 1733 y Fq(\025)3431 1741 y Fz(n)3390 1788 y Fq(n)3478 1766 y Fv(.)40 b(Then)75 1874 y Ft(g)s Fv(\()p Ft(E)223 1888 y Fq(i)252 1874 y Fv(\))26 b(=)f Ft(E)476 1888 y Fq(i)504 1874 y Fv([2)p Ft(n\025)682 1888 y Fq(i)711 1874 y Fv(])30 b(b)m(y)g(prop)s(osition)1370 1874 y SDict begin H.S end 1370 1874 a Fv(2.1)1487 1815 y SDict begin H.R end 1487 1815 a 1487 1874 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 1487 1874 a Fv(.)40 b(Th)m(us,)30 b Ft(g)f Fv(=)c(id)k(implies)e Ft(\025)2443 1888 y Fp(1)2508 1874 y Fv(=)e Fr(\001)15 b(\001)g(\001)26 b Fv(=)f Ft(\025)2884 1888 y Fq(n)2956 1874 y Fv(=)g(0.)557 b Fj(\003)75 2051 y SDict begin H.S end 75 2051 a 75 2051 a SDict begin 13 H.A end 75 2051 a 75 2051 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.5) cvn /DEST pdfmark end 75 2051 a FF(Lemma)36 b(2.5.)44 b Fl(L)-5 b(et)35 b Ft(X)43 b Fl(b)-5 b(e)35 b(a)g(smo)-5 b(oth)38 b(variety,)e Ft(T)43 b Fr(2)29 b Fv(Aut\(D)2218 2015 y Fq(b)2252 2051 y Fv(\()p Ft(X)7 b Fv(\)\))p Fl(,)37 b(and)g Ft(A;)15 b(B)35 b Fr(2)29 b Fv(D)3056 2015 y Fq(b)3090 2051 y Fv(\()p Ft(X)7 b Fv(\))36 b Fl(obje)-5 b(cts)36 b(such)75 2159 y(that)e Ft(T)13 b(A)25 b Fv(=)g Ft(A)p Fv([)p Ft(i)p Fv(])34 b Fl(and)f Ft(T)13 b(B)30 b Fv(=)25 b Ft(B)5 b Fv([)p Ft(j)g Fv(])33 b Fl(for)g(some)g Ft(i)26 b Fr(6)p Fv(=)f Ft(j)31 b Fr(2)25 b Fs(Z)p Fl(.)41 b(Then)33 b Ft(A)26 b Fr(?)f Ft(B)37 b Fl(and)c Ft(B)d Fr(?)25 b Ft(A)p Fl(.)75 2331 y(Pr)-5 b(o)g(of.)43 b Fv(See)31 b([)543 2331 y SDict begin H.S end 543 2331 a Fv(Add11)803 2268 y SDict begin H.R end 803 2268 a 803 2331 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Add) cvn H.B /ANN pdfmark end 803 2331 a Fv(,)f(p.)41 b(11].)2564 b Fj(\003)75 2503 y SDict begin H.S end 75 2503 a 75 2503 a SDict begin 13 H.A end 75 2503 a 75 2503 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.6) cvn /DEST pdfmark end 75 2503 a FF(Remark)23 b(2.6.)35 b Fv(This)19 b(sho)m(ws)h(together)i(with)e(prop)s(osition)2122 2503 y SDict begin H.S end 2122 2503 a Fv(2.1)2239 2445 y SDict begin H.R end 2239 2445 a 2239 2503 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 2239 2503 a 20 w Fv(that)i(for)e(a)i Fs(P)2714 2470 y Fq(n)2761 2503 y Fv(-functor)e Ft(F)34 b Fv(with)20 b Ft(H)32 b Fv(=)25 b([)p Fr(\000)p Ft(`)p Fv(])75 2611 y(for)31 b(some)g Ft(`)c Fr(2)f Fs(Z)31 b Fv(there)g(do)s(es)g(not)g (exist)g(a)h(non)e(zero-ob)5 b(ject)33 b Ft(A)f Fv(with)e Ft(T)2591 2625 y Fq(F)2649 2611 y Fv(\()p Ft(A)p Fv(\))e(=)e Ft(A)p Fv([)p Ft(i)p Fv(])32 b(for)f(an)m(y)g(v)-5 b(alues)31 b(of)75 2719 y Ft(i)k Fv(b)s(esides)f(0)h(and)g Fr(\000)p Ft(n`)22 b Fv(+)h(2)36 b(b)s(ecause)f(suc)m(h)g(an)g(ob)5 b(ject)36 b(w)m(ould)e(b)s(e)g(orthogonal)i(to)g(the)f(spanning)e (class)75 2827 y(im)14 b Ft(F)33 b Fr([)20 b Fv(k)m(er)c Ft(R)q Fv(.)75 2954 y SDict begin H.S end 75 2954 a 75 2954 a SDict begin 13 H.A end 75 2954 a 75 2954 a SDict begin [/View [/XYZ H.V]/Dest (section.3) cvn /DEST pdfmark end 75 2954 a 1286 3062 a Fv(3.)47 b Fu(The)34 b(dia)n(gonal)f (embedding)175 3224 y Fv(Let)24 b Ft(X)31 b Fv(b)s(e)23 b(a)h(smo)s(oth)f(pro)5 b(jectiv)m(e)24 b(surface)g(o)m(v)m(er)h Fs(C)f Fv(and)f(2)i Fr(\024)g Ft(n)g Fr(2)g Fs(N)p Fv(.)38 b(W)-8 b(e)25 b(denote)f(b)m(y)f Ft(\016)14 b Fv(:)31 b Ft(X)i Fr(!)25 b Ft(X)3553 3191 y Fq(n)3624 3224 y Fv(the)75 3332 y(diagonal)i(em)m(b)s(edding.)38 b(W)-8 b(e)28 b(w)m(an)m(t)g(to)h(sho)m(w)e(that)h Ft(F)23 b Fv(:)46 b(D)2048 3296 y Fq(b)2082 3332 y Fv(\()p Ft(X)7 b Fv(\))27 b Fr(!)e Fv(D)2446 3296 y Fq(b)2446 3356 y Fn(S)2501 3364 y Fz(n)2548 3332 y Fv(\()p Ft(X)2665 3299 y Fq(n)2713 3332 y Fv(\))j(giv)m(en)f(as)h(the)f(comp)s(osition)1241 3536 y(D)1310 3498 y Fq(b)1345 3536 y Fv(\()p Ft(X)7 b Fv(\))1545 3484 y Fp(triv)1523 3536 y Fr(\000)-32 b(\000)f(!)25 b Fv(D)1785 3498 y Fq(b)1785 3558 y Fn(S)1840 3566 y Fz(n)1887 3536 y Fv(\()p Ft(X)7 b Fv(\))2087 3484 y Fq(\016)2118 3492 y Fy(\003)2066 3536 y Fr(\000)-40 b(!)25 b Fv(D)2282 3498 y Fq(b)2282 3558 y Fn(S)2337 3566 y Fz(n)2384 3536 y Fv(\()p Ft(X)2501 3498 y Fq(n)2549 3536 y Fv(\))75 3699 y(of)43 b(the)h(functor)e(whic)m(h)g(maps)h(eac)m(h)h(ob)5 b(ject)44 b(to)g(itself)e(equipp)s(ed)e(with)i(the)i(trivial)d(action)i (and)g(the)75 3807 y(equiv)-5 b(arian)m(t)29 b(push-forw)m(ard)g(is)g (a)i Fs(P)1337 3774 y Fq(n)p Fo(\000)p Fp(1)1475 3807 y Fv(-functor.)40 b(Its)30 b(righ)m(t)g(adjoin)m(t)g Ft(R)h Fv(is)f(giv)m(en)g(as)h(the)f(comp)s(osition)1213 4016 y(D)1282 3978 y Fq(b)1282 4039 y Fn(S)1337 4047 y Fz(n)1384 4016 y Fv(\()p Ft(X)1501 3978 y Fq(n)1549 4016 y Fv(\))1631 3964 y Fq(\016)1664 3941 y Fx(!)1609 4016 y Fr(\000)-51 b(!)25 b Fv(D)1814 3978 y Fq(b)1814 4039 y Fn(S)1869 4047 y Fz(n)1916 4016 y Fv(\()p Ft(X)7 b Fv(\))2116 3956 y Fp([)p 2136 3968 48 3 v 47 w(])2203 3932 y Fi(S)2251 3940 y Fz(n)2095 4016 y Fr(\000)-24 b(\000)j(\000)d(!)25 b Fv(D)2425 3978 y Fq(b)2459 4016 y Fv(\()p Ft(X)7 b Fv(\))75 4180 y(of)33 b(the)g(usual)f(righ)m(t)h (adjoin)m(t)f(\(see)i([)1324 4180 y SDict begin H.S end 1324 4180 a Fv(LH09)1540 4117 y SDict begin H.R end 1540 4117 a 1540 4180 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Has) cvn H.B /ANN pdfmark end 1540 4180 a Fv(])g(for)e(equiv)-5 b(arian)m(t)33 b(Grothendiec)m(k)g(dualit)m(y\))f(and)g(the)h(functor) 75 4287 y(of)e(taking)g(in)m(v)-5 b(arian)m(ts.)41 b(W)-8 b(e)32 b(consider)d(the)i(standard)f(represen)m(tation)h Ft(\045)g Fv(of)g Fm(S)2814 4301 y Fq(n)2892 4287 y Fv(as)g(the)g (quotien)m(t)g(of)g(the)75 4395 y(regular)38 b(represen)m(tation)h Fs(C)1070 4362 y Fq(n)1156 4395 y Fv(b)m(y)f(the)h(one)h(dimensional)c (in)m(v)-5 b(arian)m(t)38 b(subspace.)65 b(The)39 b(normal)e(bundle)75 4503 y(sequence)1360 4629 y(0)25 b Fr(!)g Ft(T)1599 4643 y Fq(X)1692 4629 y Fr(!)g Ft(T)1861 4648 y Fq(X)1924 4629 y Fz(n)1967 4648 y Fo(j)p Fq(X)2079 4629 y Fr(!)h Ft(N)35 b Fr(!)25 b Fv(0)75 4774 y(where)30 b Ft(N)35 b Fv(:=)25 b Ft(N)640 4789 y Fq(\016)703 4774 y Fv(=)g Ft(N)872 4792 y Fq(X)o(=X)1027 4774 y Fz(n)40 b Fv(is)30 b(of)g(the)h(form)1399 4951 y(0)25 b Fr(!)h Ft(T)1639 4965 y Fq(X)1731 4951 y Fr(!)f Ft(T)1913 4913 y Fo(\010)p Fq(n)1900 4980 y(X)2040 4951 y Fr(!)g Ft(N)36 b Fr(!)25 b Fv(0)75 5122 y(where)37 b(the)h(map)f Ft(T)771 5136 y Fq(X)875 5122 y Fr(!)g Ft(T)1069 5084 y Fo(\010)p Fq(n)1056 5151 y(X)1208 5122 y Fv(is)g(the)h(diagonal)e(em)m(b)s(edding.)61 b(When)37 b(considering)f Ft(T)3166 5141 y Fq(X)3229 5122 y Fz(n)3271 5141 y Fo(j)p Fq(X)3396 5122 y Fv(as)i(a)f Fm(S)3672 5136 y Fq(n)3720 5122 y Fv(-)75 5230 y(sheaf)30 b(equipp)s(ed)d(with)h(the)i(natural)f(linearisation)e(it)i(is)g(giv)m (en)h(b)m(y)f Ft(T)2484 5244 y Fq(X)2570 5230 y Fr(\012)19 b Fs(C)2735 5197 y Fq(n)2811 5230 y Fv(where)30 b Fs(C)3149 5197 y Fq(n)3225 5230 y Fv(is)f(the)h(regular)p eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 67 -8 a SDict begin H.S end 67 -8 a 67 -8 a SDict begin H.R end 67 -8 a 67 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.5) cvn /DEST pdfmark end 67 -8 a 329 66 a Fp(NEW)30 b(DERIVED)h(A)n(UTOEQUIV)-8 b(ALENCES)30 b(OF)h(HILBER)-6 b(T)31 b(SCHEMES)f(AND)g(KUMMER)g(V)-8 b(ARIETIES)219 b(5)75 266 y Fv(represen)m(tation.)41 b(Th)m(us,)30 b(as)h(a)f Fm(S)1223 280 y Fq(n)1271 266 y Fv(-sheaf,)h(the)f(normal)g(bundle)e (equals)i Ft(T)2649 280 y Fq(X)2737 266 y Fr(\012)20 b Ft(\045)p Fv(.)40 b(W)-8 b(e)32 b(also)f(see)g(that)g(the)75 374 y(normal)e(bundle)f(sequence)j(splits)d(using)h(e.g.)j(the)f (splitting)855 572 y Ft(T)908 586 y Fq(X)996 572 y Fr(\012)20 b Fs(C)1162 534 y Fq(n)1234 572 y Fr(!)25 b Ft(T)1403 586 y Fq(X)1561 572 y Ft(;)107 b Fv(\()p Ft(v)1772 586 y Fp(1)1812 572 y Ft(;)15 b(:)g(:)g(:)h(;)f(v)2057 586 y Fq(n)2105 572 y Fv(\))25 b Fr(7!)2296 510 y Fv(1)p 2291 551 55 4 v 2291 634 a Ft(n)2356 572 y Fv(\()p Ft(v)2435 586 y Fp(1)2495 572 y Fv(+)20 b Fr(\001)15 b(\001)g(\001)21 b Fv(+)f Ft(v)2847 586 y Fq(n)2894 572 y Fv(\))15 b Ft(:)75 773 y SDict begin H.S end 75 773 a 75 773 a SDict begin 13 H.A end 75 773 a 75 773 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.1) cvn /DEST pdfmark end 75 773 a FF(Theorem)38 b(3.1)i Fv(\([)756 773 y SDict begin H.S end 756 773 a Fv(A)m(C12)979 711 y SDict begin H.R end 979 711 a 979 773 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.AC) cvn H.B /ANN pdfmark end 979 773 a Fv(]\))p FF(.)k Fl(L)-5 b(et)36 b Ft(\023)10 b Fv(:)33 b Ft(Z)38 b(,)-15 b Fr(!)31 b Ft(M)46 b Fl(b)-5 b(e)36 b(a)h(r)-5 b(e)g(gular)37 b(emb)-5 b(e)g(dding)38 b(of)e(c)-5 b(o)g(dimension)38 b Ft(c)e Fl(such)h(that)75 881 y(the)c(normal)h(bund)5 b(le)33 b(se)-5 b(quenc)g(e)32 b(splits.)43 b(Then)33 b(ther)-5 b(e)34 b(is)e(an)h(isomorphism)1280 1110 y Ft(\023)1312 1073 y Fo(\003)1352 1110 y Ft(\023)1384 1124 y Fo(\003)1424 1110 y Fv(\()p 1459 1125 61 4 v 61 w(\))25 b Fr(')g Fv(\()p 1711 1125 V 61 w(\))c Fr(\012)f Fv(\()2007 996 y Fq(c)1954 1024 y Fk(M)1966 1219 y Fq(i)p Fp(=0)2107 1110 y Fr(^)2168 1073 y Fq(i)2195 1110 y Ft(N)2278 1073 y Fo(_)2268 1137 y Fq(Z)q(=)l(M)2427 1110 y Fv([)p Ft(i)p Fv(]\))75 1110 y SDict begin H.S end 75 1110 a 75 1110 a SDict begin 16.25 H.A end 75 1110 a 75 1110 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.2) cvn /DEST pdfmark end 75 1110 a Fv(\(2\))75 1371 y Fl(of)33 b(endofunctors)h(of)f Fv(D)898 1335 y Fq(b)933 1371 y Fv(\()p Ft(Z)7 b Fv(\))p Fl(.)75 1541 y SDict begin H.S end 75 1541 a 75 1541 a SDict begin 13 H.A end 75 1541 a 75 1541 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.2) cvn /DEST pdfmark end 75 1541 a FF(Corollary)37 b(3.2.)42 b Fl(Under)32 b(the)i(same)f (assumptions,)h(ther)-5 b(e)34 b(is)f(an)g(isomorphism)1253 1770 y Ft(\023)1285 1732 y Fp(!)1309 1770 y Ft(\023)1341 1784 y Fo(\003)1381 1770 y Fv(\()p 1416 1785 61 4 v 60 w(\))26 b Fr(')f Fv(\()p 1668 1785 V 61 w(\))c Fr(\012)e Fv(\()1964 1656 y Fq(c)1910 1684 y Fk(M)1922 1879 y Fq(i)p Fp(=0)2063 1770 y Fr(^)2124 1732 y Fq(i)2152 1770 y Ft(N)2225 1789 y Fq(Z)q(=)l(M)2384 1770 y Fv([)p Fr(\000)p Ft(i)p Fv(]\))75 1770 y SDict begin H.S end 75 1770 a 75 1770 a SDict begin 16.25 H.A end 75 1770 a 75 1770 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.3) cvn /DEST pdfmark end 75 1770 a Fv(\(3\))75 2038 y Fl(Pr)-5 b(o)g(of.)43 b Fv(T)-8 b(ensorise)30 b(b)s(oth)f(sides)g(of)i(\()1325 2038 y SDict begin H.S end 1325 2038 a Fv(2)1371 1979 y SDict begin H.R end 1371 1979 a 1371 2038 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.2) cvn H.B /ANN pdfmark end 1371 2038 a Fv(\))f(b)m(y)h Ft(\023)1595 2005 y Fp(!)1619 2038 y Fr(O)1691 2052 y Fq(M)1795 2038 y Fr(')25 b(^)1952 2005 y Fq(c)1987 2038 y Ft(N)2060 2056 y Fq(Z)q(=)l(M)2219 2038 y Fv([)p Fr(\000)p Ft(c)p Fv(].)1275 b Fj(\003)175 2214 y Fv(In)29 b(the)i(case)g(that)g Ft(\023)26 b Fv(=)f Ft(\016)34 b Fv(from)c(ab)s(o)m(v)m(e)h(this)f(yields)e(the)j (isomorphism)c(of)k(functors)1121 2473 y Ft(\016)1164 2436 y Fp(!)1189 2473 y Ft(\016)1229 2487 y Fo(\003)1269 2473 y Fv(\()p 1304 2488 61 4 v 61 w(\))25 b Fr(')g Fv(\()p 1556 2488 V 61 w(\))c Fr(\012)f Fv(\()1799 2351 y Fp(2\()p Fq(n)p Fo(\000)p Fp(1\))1842 2387 y Fk(M)1853 2583 y Fq(i)p Fp(=0)2037 2473 y Fr(^)2098 2436 y Fq(i)2126 2473 y Fv(\()p Ft(T)2214 2487 y Fq(X)2302 2473 y Fr(\012)g Ft(\045)p Fv(\)[)p Fr(\000)p Ft(i)p Fv(]\))15 b Ft(:)75 2473 y SDict begin H.S end 75 2473 a 75 2473 a SDict begin 16.25 H.A end 75 2473 a 75 2473 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.4) cvn /DEST pdfmark end 75 2473 a Fv(\(4\))75 2725 y SDict begin H.S end 75 2725 a 75 2725 a SDict begin 13 H.A end 75 2725 a 75 2725 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.3) cvn /DEST pdfmark end 75 2725 a FF(Lemma)33 b(3.3.)42 b Fl(F)-7 b(or)34 b Fv(0)25 b Fr(\024)g Ft(i;)15 b(j)32 b Fr(\024)25 b Ft(c)33 b Fl(with)g Ft(i)21 b Fv(+)f Ft(j)30 b Fr(\024)25 b Ft(c)33 b Fl(the)g(c)-5 b(omp)g(onent)1015 2885 y Fv(\()p 1050 2900 61 4 v 61 w(\))20 b Fr(\012)g(^)1318 2848 y Fq(i)1346 2885 y Ft(N)1419 2904 y Fq(Z)q(=)l(M)1598 2885 y Fr(\012)g(^)1750 2848 y Fq(j)1786 2885 y Ft(N)1859 2904 y Fq(Z)q(=)l(M)2044 2885 y Fr(!)25 b Fv(\()p 2195 2900 V 61 w(\))20 b Fr(\012)g(^)2463 2848 y Fq(i)p Fp(+)p Fq(j)2578 2885 y Ft(N)2651 2904 y Fq(Z)q(=)l(M)75 3060 y Fl(of)33 b(the)g(monad)h(multiplic)-5 b(ation)35 b Ft(\023)1236 3027 y Fp(!)1260 3060 y Ft("\023)1334 3074 y Fo(\003)1384 3060 y Fv(:)c Ft(\023)1472 3027 y Fp(!)1496 3060 y Ft(\023)1528 3074 y Fo(\003)1567 3060 y Ft(\023)1599 3027 y Fp(!)1623 3060 y Ft(\023)1655 3074 y Fo(\003)1720 3060 y Fr(!)25 b Ft(\023)1868 3027 y Fp(!)1892 3060 y Ft(\023)1924 3074 y Fo(\003)1996 3060 y Fl(is)33 b(given)f(by)g(the)h (we)-5 b(dge)33 b(p)-5 b(airing.)75 3230 y(Pr)g(o)g(of.)43 b Fv(F)-8 b(or)31 b Ft(E)f Fr(2)25 b Fv(D)771 3194 y Fq(b)806 3230 y Fv(\()p Ft(M)10 b Fv(\))30 b(the)h(ob)5 b(ject)30 b Ft(\023)1467 3197 y Fp(!)1491 3230 y Ft(E)36 b Fv(can)30 b(b)s(e)f(iden)m(ti\014ed)f(with)h Fr(H)q Fl(om)p Fv(\()p Ft(\023)2751 3244 y Fo(\003)2791 3230 y Fr(O)2863 3244 y Fq(Z)2920 3230 y Ft(;)15 b(E)5 b Fv(\))31 b(considered)e(as)h(an)75 3343 y(ob)5 b(ject)26 b(in)f(D)516 3307 y Fq(b)550 3343 y Fv(\()p Ft(Z)7 b Fv(\).)40 b(Under)24 b(this)h(iden)m(ti\014cation)f(the)h(counit)g(map)h Fr(H)q Fl(om)p Fv(\()p Ft(\023)2627 3357 y Fo(\003)2667 3343 y Fr(O)2739 3357 y Fq(Z)2796 3343 y Ft(;)15 b(E)5 b Fv(\))26 b Fr(!)g Ft(E)k Fv(is)25 b(giv)m(en)g(b)m(y)h(the)75 3456 y(ev)-5 b(aluation)23 b Ft(')j Fr(7!)f Ft(')p Fv(\(1\))h(\(see)e ([)1106 3456 y SDict begin H.S end 1106 3456 a Fv(Har66)1347 3394 y SDict begin H.R end 1347 3394 a 1347 3456 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Har1) cvn H.B /ANN pdfmark end 1347 3456 a Fv(,)h(section)f(I)s(I)s(I.6]\).)38 b(No)m(w)25 b(w)m(e)f(get)h(for)f Ft(F)38 b Fr(2)25 b Fv(D)2849 3420 y Fq(b)2883 3456 y Fv(\()p Ft(Z)7 b Fv(\))24 b(the)g(iden)m(ti\014cations)722 3623 y Ft(\023)754 3585 y Fp(!)778 3623 y Ft(\023)810 3637 y Fo(\003)849 3623 y Ft(\023)881 3585 y Fp(!)905 3623 y Ft(\023)937 3637 y Fo(\003)977 3623 y Ft(F)38 b Fr(')25 b(H)q Fl(om)q Fv(\()p Ft(\023)1435 3637 y Fo(\003)1475 3623 y Fr(O)1547 3637 y Fq(Z)1604 3623 y Ft(;)15 b(\023)1676 3637 y Fo(\003)1716 3623 y Fr(O)1788 3637 y Fq(Z)1845 3623 y Fv(\))20 b Fr(\012)1971 3637 y Fo(O)2027 3648 y Fz(Z)2101 3623 y Fr(H)q Fl(om)p Fv(\()p Ft(\023)2366 3637 y Fo(\003)2406 3623 y Fr(O)2478 3637 y Fq(Z)2535 3623 y Ft(;)15 b(\023)2607 3637 y Fo(\003)2647 3623 y Fr(O)2719 3637 y Fq(Z)2776 3623 y Fv(\))21 b Fr(\012)2903 3637 y Fo(O)2959 3648 y Fz(Z)3032 3623 y Ft(F)75 3787 y Fv(and)30 b Ft(\023)284 3754 y Fp(!)308 3787 y Ft(\023)340 3801 y Fo(\003)379 3787 y Ft(F)39 b Fr(')25 b(H)q Fl(om)p Fv(\()p Ft(\023)837 3801 y Fo(\003)877 3787 y Fr(O)949 3801 y Fq(Z)1006 3787 y Ft(;)15 b(\023)1078 3801 y Fo(\003)1118 3787 y Fr(O)1190 3801 y Fq(Z)1247 3787 y Fv(\))21 b Fr(\012)1374 3801 y Fo(O)1430 3812 y Fz(Z)1503 3787 y Ft(F)13 b Fv(.)41 b(Under)29 b(these)i(iden)m(ti\014cations)e(the)h(comp)s(onen)m(t)389 3954 y Fr(E)8 b Fl(xt)517 3917 y Fq(i)545 3954 y Fv(\()p Ft(\023)612 3968 y Fo(\003)652 3954 y Fr(O)724 3968 y Fq(Z)781 3954 y Ft(;)15 b(\023)853 3968 y Fo(\003)893 3954 y Fr(O)965 3968 y Fq(Z)1022 3954 y Fv(\))21 b Fr(\012)1149 3968 y Fo(O)1205 3979 y Fz(Z)1278 3954 y Fr(E)8 b Fl(xt)1407 3917 y Fq(j)1443 3954 y Fv(\()p Ft(\023)1510 3968 y Fo(\003)1550 3954 y Fr(O)1622 3968 y Fq(Z)1679 3954 y Ft(;)15 b(\023)1751 3968 y Fo(\003)1791 3954 y Fr(O)1863 3968 y Fq(Z)1920 3954 y Fv(\))21 b Fr(\012)2047 3968 y Fo(O)2103 3979 y Fz(Z)2176 3954 y Ft(F)38 b Fr(!)26 b(E)8 b Fl(xt)2517 3917 y Fq(i)p Fp(+)p Fq(j)2632 3954 y Fv(\()p Ft(\023)2699 3968 y Fo(\003)2739 3954 y Fr(O)2811 3968 y Fq(Z)2868 3954 y Ft(;)15 b(\023)2940 3968 y Fo(\003)2980 3954 y Fr(O)3052 3968 y Fq(Z)3110 3954 y Fv(\))20 b Fr(\012)3236 3968 y Fo(O)3292 3979 y Fz(Z)3365 3954 y Ft(F)75 4114 y Fv(of)28 b(the)g(monad)f(m)m(ultiplication)e(equals)i(the)h(Y)-8 b(oneda)29 b(pro)s(duct.)38 b(The)28 b(Y)-8 b(oneda)28 b(pro)s(duct)f(corresp)s(onds)f(to)75 4222 y(the)j(w)m(edge)g(pro)s (duct)f(under)f(the)i(isomorphism)c Fr(E)8 b Fl(xt)1907 4187 y Fq(i)1935 4222 y Fv(\()p Fr(O)2042 4236 y Fq(Z)2099 4222 y Ft(;)15 b Fr(O)2211 4236 y Fq(Z)2269 4222 y Fv(\))2329 4197 y Fr(\030)2329 4227 y Fv(=)2425 4222 y Fr(^)2486 4189 y Fq(i)2514 4222 y Ft(N)2587 4241 y Fq(Z)q(=)l(M)2775 4222 y Fv(\(see)29 b([)2980 4222 y SDict begin H.S end 2980 4222 a Fv(LH09)3196 4160 y SDict begin H.R end 3196 4160 a 3196 4222 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Has) cvn H.B /ANN pdfmark end 3196 4222 a Fv(,)h(p.)39 b(442]\).)93 b Fj(\003)75 4396 y SDict begin H.S end 75 4396 a 75 4396 a SDict begin 13 H.A end 75 4396 a 75 4396 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.4) cvn /DEST pdfmark end 75 4396 a FF(Lemma)36 b(3.4)j Fv(\([)681 4396 y SDict begin H.S end 681 4396 a Fv(Sca09a)954 4334 y SDict begin H.R end 954 4334 a 954 4396 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Sca1) cvn H.B /ANN pdfmark end 954 4396 a Fv(,)34 b(App)s(endix)d(B]\))p FF(.)44 b Fl(L)-5 b(et)35 b Ft(V)56 b Fl(b)-5 b(e)35 b(a)g(two-dimensional)j (ve)-5 b(ctor)36 b(sp)-5 b(ac)g(e)36 b(with)h(a)e(b)-5 b(asis)75 4506 y(c)g(onsisting)32 b(of)g(ve)-5 b(ctors)33 b Ft(u)e Fl(and)i Ft(v)s Fl(.)41 b(Then)32 b(the)g(sp)-5 b(ac)g(e)33 b(of)f(invariants)h Fv([)p Fr(^)2524 4473 y Fq(i)2552 4506 y Fv(\()p Ft(V)38 b Fr(\012)18 b Ft(\045)p Fv(\)])2874 4473 y Fn(S)2929 4481 y Fz(n)3008 4506 y Fl(is)31 b(one-dimensional)75 4614 y(if)d Fv(0)e Fr(\024)f Ft(i)h Fr(\024)f Fv(2\()p Ft(n)12 b Fr(\000)g Fv(1\))29 b Fl(is)g(even)f(and)h(zer)-5 b(o)30 b(if)f(it)f(is)h(o)-5 b(dd.)42 b(In)29 b(the)g(even)f(c)-5 b(ase)30 b Ft(i)25 b Fv(=)g(2)p Ft(`)k Fl(the)g(sp)-5 b(ac)g(e)31 b(of)e(invariants)75 4724 y(is)k(sp)-5 b(anne)g(d)34 b(by)f(the)g(image)g(of)g(the)g(ve)-5 b(ctor)33 b Ft(!)1637 4691 y Fq(`)1670 4724 y Fl(,)f(wher)-5 b(e)1264 4971 y Ft(!)28 b Fv(=)1491 4857 y Fq(k)1445 4885 y Fk(X)1453 5080 y Fq(i)p Fp(=1)1591 4971 y Ft(ue)1685 4985 y Fq(i)1734 4971 y Fr(^)20 b Ft(v)s(e)1904 4985 y Fq(i)1958 4971 y Fr(2)25 b(^)2105 4934 y Fp(2)2144 4971 y Fv(\()p Ft(V)41 b Fr(\012)20 b Fs(C)2439 4934 y Fq(n)2485 4971 y Fv(\))15 b Ft(;)75 5213 y Fl(under)33 b(the)g(pr)-5 b(oje)g(ction)35 b(induc)-5 b(e)g(d)33 b(by)g(the)g(pr)-5 b(oje)g(ction)34 b Fs(C)2000 5180 y Fq(n)2073 5213 y Fr(!)25 b Ft(\045)p Fl(.)p eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 67 -8 a SDict begin H.S end 67 -8 a 67 -8 a SDict begin H.R end 67 -8 a 67 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.6) cvn /DEST pdfmark end 67 -8 a 75 66 a Fp(6)1508 b(ANDREAS)24 b(KR)n(UG)75 266 y SDict begin H.S end 75 266 a 75 266 a SDict begin 13 H.A end 75 266 a 75 266 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.5) cvn /DEST pdfmark end 75 266 a FF(Corollary)47 b(3.5.)g Fl(F)-7 b(or)41 b(a)g(ve)-5 b(ctor)42 b(bund)5 b(le)40 b Ft(E)46 b Fl(on)41 b Ft(X)48 b Fl(of)41 b(r)-5 b(ank)42 b(two)f(and)h Fv(0)e Fr(\024)g Ft(`)f Fr(\024)h Ft(n)26 b Fr(\000)f Fv(1)41 b Fl(ther)-5 b(e)42 b(is)e(an)75 374 y(isomorphism)1375 488 y Fv([)p Fr(^)1461 450 y Fp(2)p Fq(`)1529 488 y Fv(\()p Ft(E)26 b Fr(\012)20 b Ft(\045)p Fv(\)])1855 450 y Fn(S)1910 458 y Fz(n)1983 463 y Fr(\030)1983 492 y Fv(=)2079 488 y(\()p Fr(^)2175 450 y Fp(2)2214 488 y Ft(E)5 b Fv(\))2321 450 y Fo(\012)p Fq(`)2425 488 y Ft(:)75 652 y Fl(Pr)-5 b(o)g(of.)43 b Fv(The)30 b(isomorphism)d(is)j(giv)m(en)g(b)m(y)g(comp)s (osing)g(the)g(morphism)420 816 y(\()p Fr(^)516 778 y Fp(2)556 816 y Ft(E)5 b Fv(\))663 778 y Fo(\012)p Fq(`)776 816 y Fr(!)26 b(^)954 778 y Fq(`)986 816 y Fv(\()p Ft(E)g Fr(\012)20 b Fs(C)1280 778 y Fq(n)1327 816 y Fv(\))91 b Ft(;)107 b(x)1637 830 y Fp(1)1696 816 y Fr(\012)20 b(\001)15 b(\001)g(\001)21 b(\012)f Ft(x)2056 831 y Fq(`)2114 816 y Fr(7!)2399 730 y Fk(X)2231 926 y Fp(1)p Fo(\024)p Fq(i)2345 935 y Fx(1)2379 926 y Fq(<)p Fo(\001\001\001)o Fq(<i)2572 938 y Fz(`)2602 926 y Fo(\024)p Fq(n)2715 816 y Ft(x)2767 830 y Fp(1)2806 816 y Ft(e)2848 830 y Fq(i)2872 839 y Fx(1)2931 816 y Fr(^)g(\001)15 b(\001)g(\001)22 b(^)d Ft(x)3271 831 y Fq(`)3304 816 y Ft(e)3346 830 y Fq(i)3370 842 y Fz(`)75 1060 y Fv(with)29 b(the)i(pro)5 b(jection)30 b(induced)e(b)m(y)i(the)h(pro)5 b(jection)30 b Fs(C)1999 1027 y Fq(n)2071 1060 y Fr(!)25 b Ft(\045)p Fv(.)1420 b Fj(\003)175 1224 y Fv(W)-8 b(e)31 b(set)g Ft(H)h Fv(:=)26 b Fr(^)764 1191 y Fp(2)803 1224 y Ft(T)856 1238 y Fq(X)923 1224 y Fv([)p Fr(\000)p Fv(2])g(=)f Ft(!)1271 1191 y Fo(_)1268 1251 y Fq(X)1335 1224 y Fv([)p Fr(\000)p Fv(2])h(=)f Ft(S)1684 1186 y Fo(\000)p Fp(1)1679 1252 y Fq(X)1808 1224 y Fv(as)31 b(the)g(in)m(v)m(erse)f(of)g(the)h(Serre)f (functor)g(on)g Ft(X)7 b Fv(.)75 1388 y SDict begin H.S end 75 1388 a 75 1388 a SDict begin 13 H.A end 75 1388 a 75 1388 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.6) cvn /DEST pdfmark end 75 1388 a FF(Corollary)37 b(3.6.)42 b Fl(Ther)-5 b(e)33 b(is)g(the)g(isomorphism)i(of)e(functors) 1248 1539 y Ft(R)q(F)38 b Fr(')25 b Fv(id)14 b Fr(\010)p Ft(H)27 b Fr(\010)20 b Ft(H)1949 1501 y Fp(2)2008 1539 y Fr(\010)g(\001)15 b(\001)g(\001)22 b(\010)d Ft(H)2399 1501 y Fq(n)p Fo(\000)p Fp(1)2552 1539 y Ft(:)75 1703 y Fl(Pr)-5 b(o)g(of.)43 b Fv(This)28 b(follo)m(ws)i(b)m(y)g(form)m(ula) g(\()1357 1703 y SDict begin H.S end 1357 1703 a Fv(4)1403 1645 y SDict begin H.R end 1403 1645 a 1403 1703 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.4) cvn H.B /ANN pdfmark end 1403 1703 a Fv(\))g(and)g(corollary)2022 1703 y SDict begin H.S end 2022 1703 a Fv(3.5)2138 1645 y SDict begin H.R end 2138 1645 a 2138 1703 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.5) cvn H.B /ANN pdfmark end 2138 1703 a Fv(.)1516 b Fj(\003)75 1868 y SDict begin H.S end 75 1868 a 75 1868 a SDict begin 13 H.A end 75 1868 a 75 1868 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.7) cvn /DEST pdfmark end 75 1868 a FF(Lemma)34 b(3.7.)43 b Fl(The)34 b(map)i Ft(H)7 b(R)q(F)40 b Fr(!)28 b Ft(R)q(F)46 b Fl(de\014ne)-5 b(d)35 b(in)f(c)-5 b(ondition)36 b(\(ii\))e(for)g Fs(P)p Fl(-functors)h(is)e(for)i(this)g(p)-5 b(air)75 1976 y Ft(F)38 b Fj(\012)25 b Ft(R)34 b Fl(given)e(by)g(a)h (matrix)h(of)f(the)g(form)h(\()1631 1976 y SDict begin H.S end 1631 1976 a Fl(1)1677 1917 y SDict begin H.R end 1677 1917 a 1677 1976 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.1) cvn H.B /ANN pdfmark end 1677 1976 a Fl(\).)75 2142 y(Pr)-5 b(o)g(of.)43 b Fv(The)33 b(generators)i Ft(!)1050 2109 y Fq(`)1116 2142 y Fv(from)e(lemma)1631 2142 y SDict begin H.S end 1631 2142 a Fv(3.4)1747 2084 y SDict begin H.R end 1747 2084 a 1747 2142 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.4) cvn H.B /ANN pdfmark end 1747 2142 a 33 w Fv(are)i(mapp)s(ed)d(to)i(eac)m(h)h(other)f(b)m (y)g(w)m(edge)g(pro)s(duct.)50 b(By)75 2250 y(lemma)367 2250 y SDict begin H.S end 367 2250 a Fv(3.3)483 2192 y SDict begin H.R end 483 2192 a 483 2250 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.3) cvn H.B /ANN pdfmark end 483 2250 a 30 w Fv(the)30 b(comp)s(onen)m(ts)f Ft(H)d Fr(\016)19 b Ft(H)1417 2217 y Fq(k)1485 2250 y Fr(!)25 b Ft(H)1684 2217 y Fq(k)r Fp(+1)1846 2250 y Fv(for)30 b Ft(k)e Fv(=)d(0)p Ft(;)15 b(:)g(:)g(:)i(;)e(n)k Fr(\000)f Fv(2)30 b(of)g(the)g(monad)f(m)m(ultiplication)75 2358 y(are)i(giv)m(en)f(b)m(y)g(w)m(edge)i(pro)s(duct.)39 b(Hence,)32 b(they)e(are)h(isomorphisms.)1225 b Fj(\003)75 2523 y SDict begin H.S end 75 2523 a 75 2523 a SDict begin 13 H.A end 75 2523 a 75 2523 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.8) cvn /DEST pdfmark end 75 2523 a FF(Lemma)33 b(3.8.)42 b Fl(Ther)-5 b(e)33 b(is)g(the)g(isomorphism)i Ft(S)1751 2537 y Fq(X)1814 2518 y Fz(n)1861 2523 y Ft(F)13 b(H)2015 2490 y Fq(n)p Fo(\000)p Fp(1)2177 2523 y Fr(')25 b Ft(F)13 b(S)2400 2537 y Fq(X)2467 2523 y Fl(.)75 2687 y(Pr)-5 b(o)g(of.)43 b Fv(F)-8 b(or)31 b Fr(E)i(2)25 b Fv(D)756 2651 y Fq(b)790 2687 y Fv(\()p Ft(X)7 b Fv(\))31 b(there)g(are)g(natural)e(isomorphisms)227 2858 y Ft(S)283 2872 y Fq(X)346 2853 y Fz(n)393 2858 y Ft(F)13 b(H)547 2821 y Fq(n)p Fo(\000)p Fp(1)684 2858 y Fv(\()p Fr(E)8 b Fv(\))26 b(=)f Ft(!)989 2872 y Fq(X)1052 2853 y Fz(n)1098 2858 y Fv([2)p Ft(n)p Fv(])c Fr(\012)f Ft(\016)1400 2872 y Fo(\003)1440 2858 y Fv(\()p Fr(E)29 b(\012)19 b Ft(!)1702 2810 y Fo(\000)p Fp(\()p Fq(n)p Fo(\000)p Fp(1\))1699 2887 y Fq(X)1949 2858 y Fv([)p Fr(\000)p Fv(2\()p Ft(n)i Fr(\000)f Fv(1\)]\))26 b Fr(')f Ft(!)2614 2821 y Fh(\002)p Fq(n)2611 2881 y(X)2733 2858 y Fr(\012)20 b Ft(\016)2864 2872 y Fo(\003)2904 2858 y Fv(\()p Fr(E)28 b(\012)20 b Ft(!)3166 2810 y Fo(\000)p Fp(\()p Fq(n)p Fo(\000)p Fp(1\))3163 2887 y Fq(X)3413 2858 y Fv(\)[2])2458 2984 y Fp(PF)2470 3045 y Fr(')37 b Ft(\016)2618 3059 y Fo(\003)2657 3045 y Fv(\()p Fr(E)29 b(\012)20 b Ft(!)2917 3059 y Fq(X)2984 3045 y Fv([2]\))27 b(=)d Ft(F)13 b(S)3363 3059 y Fq(X)3431 3045 y Fv(\()p Fr(E)8 b Fv(\))15 b Ft(:)3679 3195 y Fj(\003)175 3359 y Fv(All)29 b(this)g(together)j(sho)m(ws)e(theorem)1475 3359 y SDict begin H.S end 1475 3359 a Fv(1.1)1591 3300 y SDict begin H.R end 1591 3300 a 1591 3359 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.1) cvn H.B /ANN pdfmark end 1591 3359 a Fv(,)g(i.e.)h(that)g Ft(F)38 b Fv(=)25 b Ft(\016)2222 3373 y Fo(\003)2282 3359 y Fr(\016)c Fv(triv)31 b(is)e(indeed)g(a)i Fs(P)3050 3326 y Fq(n)p Fo(\000)p Fp(1)3187 3359 y Fv(-functor.)75 3465 y SDict begin H.S end 75 3465 a 75 3465 a SDict begin 13 H.A end 75 3465 a 75 3465 a SDict begin [/View [/XYZ H.V]/Dest (section.4) cvn /DEST pdfmark end 75 3465 a 315 3555 a Fv(4.)46 b Fu(Composition)34 b(with)g(the)g(Bridgeland{King{Reid{Haiman)f(equiv) -8 b(alence)175 3717 y Fv(The)34 b Fl(isosp)-5 b(e)g(ctr)g(al)39 b(Hilb)-5 b(ert)37 b(scheme)42 b Ft(I)1480 3684 y Fq(n)1527 3717 y Ft(X)e Fr(\032)32 b Ft(X)1827 3684 y Fp([)p Fq(n)p Fp(])1937 3717 y Fr(\002)23 b Ft(X)2113 3684 y Fq(n)2195 3717 y Fv(is)33 b(de\014ned)h(as)h(the)g(reduced)f(\014bre)f(pro)s (duct)75 3826 y Ft(I)122 3793 y Fq(n)169 3826 y Ft(X)j Fv(:=)405 3752 y Fk(\000)446 3826 y Ft(X)528 3793 y Fp([)p Fq(n)p Fp(])635 3826 y Fr(\002)706 3840 y Fq(S)753 3821 y Fz(n)795 3840 y Fq(X)883 3826 y Ft(X)965 3793 y Fq(n)1012 3752 y Fk(\001)1054 3862 y Fp(red)1188 3826 y Fv(with)31 b(the)i(de\014ning)d(morphisms)g(b)s(eing)h(the)i(Hilb)s(ert{Cho)m(w)d (morphism)75 3953 y Ft(\026)10 b Fv(:)31 b Ft(X)278 3920 y Fp([)p Fq(n)p Fp(])390 3953 y Fr(!)26 b Ft(S)568 3920 y Fq(n)615 3953 y Ft(X)39 b Fv(and)30 b(the)h(quotien)m(t)g(morphism)e Ft(\031)13 b Fv(:)31 b Ft(X)2051 3920 y Fq(n)2124 3953 y Fr(!)26 b Ft(S)2302 3920 y Fq(n)2349 3953 y Ft(X)7 b Fv(.)43 b(Th)m(us,)30 b(there)h(is)f(the)h(comm)m(utativ)m(e)75 4061 y(diagram)1539 4185 y Ft(I)1586 4153 y Fq(n)1633 4185 y Ft(X)1868 4129 y Fq(p)1761 4185 y Fr(\000)-43 b(\000)-20 b(\000)g(\000)-43 b(!)96 b Ft(X)2188 4153 y Fq(n)1563 4346 y(q)1597 4257 y Fk(?)1597 4311 y(?)1597 4366 y(y)2140 4257 y(?)2140 4311 y(?)2140 4366 y(y)2201 4353 y Fq(\031)1543 4537 y Ft(X)1625 4504 y Fp([)p Fq(n)p Fp(])1761 4537 y Fr(\000)-43 b(\000)-20 b(\000)g(\000)-43 b(!)1864 4592 y Fq(\026)2055 4537 y Ft(S)2116 4504 y Fq(n)2163 4537 y Ft(X)23 b(:)75 4709 y Fv(The)30 b Fl (Bridgeland{King{R)-5 b(eid{Haiman)37 b(e)-5 b(quivalenc)g(e)37 b Fv(is)29 b(the)i(functor)709 4864 y(\010)25 b(:=)g(FM)1064 4878 y Fo(O)1120 4891 y Fz(I)1151 4877 y(n)1193 4891 y(X)1271 4864 y Fr(\016)15 b Fv(triv)26 b(=)f Ft(p)1644 4878 y Fo(\003)1703 4864 y Fr(\016)c Ft(q)1813 4827 y Fo(\003)1872 4864 y Fr(\016)g Fv(triv)11 b(:)45 b(D)2234 4827 y Fq(b)2268 4864 y Fv(\()p Ft(X)2385 4827 y Fp([)p Fq(n)p Fp(])2472 4864 y Fv(\))26 b Fr(\000)-16 b(!)25 b Fv(D)2774 4827 y Fq(b)2774 4887 y Fn(S)2829 4895 y Fz(n)2875 4864 y Fv(\()p Ft(X)2992 4827 y Fq(n)3040 4864 y Fv(\))15 b Ft(:)75 5015 y Fv(By)36 b(the)g(results)f(in)g([)817 5015 y SDict begin H.S end 817 5015 a Fv(BKR01)1110 4952 y SDict begin H.R end 1110 4952 a 1110 5015 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.BKR) cvn H.B /ANN pdfmark end 1110 5015 a Fv(])h(and)g([)1379 5015 y SDict begin H.S end 1379 5015 a Fv(Hai01)1609 4952 y SDict begin H.R end 1609 4952 a 1609 5015 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Hai) cvn H.B /ANN pdfmark end 1609 5015 a Fv(])g(it)f(is)g(indeed)g(an)h(equiv)-5 b(alence.)57 b(The)35 b(isosp)s(ectral)g(Hilb)s(ert)75 5122 y(sc)m(heme)28 b(can)f(b)s(e)f(iden)m(ti\014ed)f(with)g(the)i(blo)m(w-up)f(of)h Ft(X)1940 5089 y Fq(n)2014 5122 y Fv(along)g(the)f(union)f(of)i(all)f (the)h(pairwise)e(diagonals)75 5230 y(\001)151 5244 y Fq(ij)246 5230 y Fv(=)34 b Fr(f)p Fv(\()p Ft(x)483 5244 y Fp(1)523 5230 y Ft(;)15 b(:)g(:)g(:)i(;)e(x)777 5244 y Fq(n)824 5230 y Fv(\))35 b Fr(2)g Ft(X)1072 5197 y Fq(n)1154 5230 y Fr(j)g Ft(x)1266 5244 y Fq(i)1328 5230 y Fv(=)g Ft(x)1486 5244 y Fq(j)1522 5230 y Fr(g)h Fv(\(see)h([)1816 5230 y SDict begin H.S end 1816 5230 a Fv(Hai01)2046 5168 y SDict begin H.R end 2046 5168 a 2046 5230 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Hai) cvn H.B /ANN pdfmark end 2046 5230 a Fv(]\).)59 b(By)36 b(lemma)2636 5230 y SDict begin H.S end 2636 5230 a Fv(2.3)2753 5172 y SDict begin H.R end 2753 5172 a 2753 5230 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.3) cvn H.B /ANN pdfmark end 2753 5230 a 36 w Fv(the)g(functor)f(comp)s(osition)p eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 67 -8 a SDict begin H.S end 67 -8 a 67 -8 a SDict begin H.R end 67 -8 a 67 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.7) cvn /DEST pdfmark end 67 -8 a 329 66 a Fp(NEW)30 b(DERIVED)h(A)n(UTOEQUIV)-8 b(ALENCES)30 b(OF)h(HILBER)-6 b(T)31 b(SCHEMES)f(AND)g(KUMMER)g(V)-8 b(ARIETIES)219 b(7)75 266 y Fv(\010)141 233 y Fo(\000)p Fp(1)257 266 y Fr(\016)23 b Ft(F)h Fv(:)46 b(D)548 230 y Fq(b)582 266 y Fv(\()p Ft(X)7 b Fv(\))32 b Fr(!)f Fv(D)957 230 y Fq(b)992 266 y Fv(\()p Ft(X)1109 233 y Fp([)p Fq(n)p Fp(])1196 266 y Fv(\))j(is)e(again)i(a)g Fs(P)1755 233 y Fq(n)p Fo(\000)p Fp(1)1893 266 y Fv(-functor)f(and)g(th)m(us)h (yields)e(an)h(auto)s(equiv)-5 b(alence)34 b(of)75 374 y(the)29 b(deriv)m(ed)e(category)j(of)f(the)g(Hilb)s(ert)d(sc)m(heme.) 41 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b(transform)g(\(see)i(e.g.)75 1515 y([)100 1533 y SDict begin H.S end 100 1533 a -18 x Fv(Huy06)358 1453 y SDict begin H.R end 358 1453 a 358 1515 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Huy) cvn H.B /ANN pdfmark end 358 1515 a Fv(,)28 b(Prop)s(osition)d(5.9]\),)30 b(w)m(e)e(ha)m(v)m (e)h Fr(Q)c Fv(=)g Fr(O)1735 1482 y Fo(_)1732 1548 y Fq(I)1768 1529 y Fx(2)1803 1548 y Fq(X)1884 1515 y Fr(\012)14 b Ft(q)2013 1482 y Fo(\003)2052 1515 y Ft(!)2109 1541 y Fq(X)2172 1521 y Fx([2])2246 1515 y Fv([4].)41 b(The)26 b(canonical)h(bundle)e(of)j(the)f(blo)m(w-)75 1624 y(up)36 b(is)h(giv)m(en)h(b)m(y)f Ft(!)747 1644 y Fq(I)783 1625 y Fx(2)817 1644 y Fq(X)922 1599 y Fr(\030)922 1628 y Fv(=)1029 1624 y Ft(p)1075 1591 y Fo(\003)1115 1624 y Ft(!)1172 1644 y Fq(X)1235 1625 y Fx(2)1298 1624 y Fr(\012)25 b(O)s Fv(\()p Ft(E)5 b Fv(\).)63 b(Let)38 b Ft(N)47 b Fv(b)s(e)37 b(the)h(normal)f(bundle)e(of)i(the)h(co)s(dimension)e(4)75 1738 y(regular)30 b(em)m(b)s(edding)e Ft(I)890 1705 y Fp(2)930 1738 y Ft(X)k Fr(!)26 b Ft(X)1236 1705 y Fp([2])1335 1738 y Fr(\002)20 b Ft(X)1508 1705 y Fp(2)1548 1738 y Fv(.)40 b(By)31 b(adjunction)e(form)m(ula)974 1907 y Fr(^)1035 1869 y Fp(4)1074 1907 y Ft(N)1182 1881 y Fr(\030)1182 1911 y Fv(=)1278 1907 y Ft(!)1338 1869 y Fo(_)1335 1940 y Fq(X)1398 1921 y Fx([2])1468 1940 y Fo(\002)p Fq(X)1586 1922 y Fx(2)1621 1940 y Fo(j)p Fq(I)1677 1922 y Fx(2)1710 1940 y Fq(X)1798 1907 y Fr(\012)20 b Ft(!)1946 1926 y Fq(I)1982 1907 y Fx(2)2016 1926 y Fq(X)2109 1881 y Fr(\030)2109 1911 y Fv(=)2205 1907 y Ft(q)2249 1869 y Fo(\003)2288 1907 y Ft(!)2348 1869 y Fo(_)2345 1940 y Fq(X)2408 1921 y Fx([2])2502 1907 y Fr(\012)g(O)s Fv(\()p Ft(E)5 b Fv(\))15 b Ft(:)75 2089 y Fv(It)30 b(follo)m(ws)g(b)m(y)g(Grothendiec)m(k-V)-8 b(erdier)30 b(dualit)m(y)f(for)i(regular)e(em)m(b)s(eddings)g(that)801 2258 y Fr(Q)c Fv(=)g Fr(O)1071 2220 y Fo(_)1068 2286 y Fq(I)1104 2267 y Fx(2)1139 2286 y Fq(X)1226 2258 y Fr(\012)20 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Fr(Q)p Fv(,)g(i.e.)f Ft(q)1867 2515 y Fo(\003)1907 2548 y Ft(q)1948 2562 y Fo(\003)1987 2548 y Fr(Q)2061 2515 y Fn(S)2116 2524 y Fx(2)2180 2548 y Fv(=)c Fr(Q)p Fv(.)42 b(W)-8 b(e)31 b(also)g(ha)m(v)m(e)h(\010)3033 2515 y Fo(\000)p Fp(1)3127 2548 y Fv(\()p Fr(O)3234 2562 y Fq(X)3297 2543 y Fz(n)3344 2548 y Fv(\))27 b(=)e Ft(q)3543 2562 y Fo(\003)3582 2548 y Fr(Q)3656 2515 y Fn(S)3711 2524 y Fx(2)75 2656 y Fv(and)33 b(b)m(y)h([)410 2656 y SDict begin H.S end 410 2656 a Fv(Sca09a)683 2594 y SDict begin H.R end 683 2594 a 683 2656 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Sca1) cvn H.B /ANN pdfmark end 683 2656 a Fv(,)i(Prop)s(osition)c(1.3.3]) k(\010)1549 2623 y Fo(\000)p Fp(1)1643 2656 y Fv(\()p Fr(O)1750 2670 y Fq(X)1813 2651 y Fz(n)1860 2656 y Fv(\))c(=)f Fr(O)2101 2682 y Fq(X)2164 2662 y Fx([)p Fz(n)p Fx(])2246 2656 y Fv(.)52 b(All)33 b(this)g(together)i(implies)c(that)k Fr(O)s Fv(\()p Ft(E)5 b Fv(\))75 2764 y(is)29 b(the)i(trivial)d(line)h (bundle)f(whic)m(h)h(is)h(a)g(con)m(tradiction.)1641 b Fj(\003)175 2952 y Fv(If)35 b(the)i(surface)f Ft(X)44 b Fv(has)35 b(trivial)g(canonical)h(bundle,)f(it)h(is)f(kno)m(wn)h (that)h(an)m(y)f(line)f(bundle)f Ft(L)i Fv(on)g(the)75 3076 y(b)s(oundary)d Ft(@)5 b(X)624 3043 y Fp([2])736 3076 y Fv(=)32 b Ft(X)921 3028 y Fp([2])914 3104 y(\001)1035 3076 y Fv(of)j(the)g(Hilb)s(ert)e(sc)m(heme)i(of)g(t)m(w)m(o)h(p)s(oin) m(ts)e(on)h Ft(X)42 b Fv(is)34 b(an)g(EZ-spherical)f(ob)5 b(ject)75 3184 y(\(see)31 b([)282 3201 y SDict begin H.S end 282 3201 a -17 x Fv(Huy06)540 3121 y SDict begin H.R end 540 3121 a 540 3184 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Huy) cvn H.B /ANN pdfmark end 540 3184 a Fv(,)g(examples)f(8.49)i(\(iv\)]\).)41 b(That)30 b(means)g(that)h (the)g(functor)1005 3339 y(~)985 3362 y Ft(F)1043 3376 y Fq(L)1105 3362 y Fv(:)46 b(D)1245 3325 y Fq(b)1280 3362 y Fv(\()p Ft(X)7 b Fv(\))26 b Fr(!)f Fv(D)1644 3325 y Fq(b)1678 3362 y Fv(\()p Ft(X)1795 3325 y Fp([2])1875 3362 y Fv(\))91 b Ft(;)106 b Fr(E)34 b(7!)25 b Ft(j)2367 3376 y Fo(\003)2407 3362 y Fv(\()p Ft(L)20 b Fr(\012)g Ft(\026)2670 3325 y Fo(\003)2670 3385 y Fp(\001)2733 3362 y Ft(E)5 b Fv(\))75 3531 y(is)29 b(a)i(spherical)e(functor)h (where)g(the)g(maps)g Ft(j)36 b Fv(and)30 b Ft(\026)1900 3545 y Fp(\001)1993 3531 y Fv(come)h(from)f(the)g(\014bre)g(diagram) 1571 3727 y Ft(X)1653 3679 y Fp([2])1646 3755 y(\001)1885 3670 y Fq(j)1777 3727 y Fr(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)56 b Ft(X)2164 3694 y Fp([2])1524 3888 y Fq(\026)1566 3899 y Fx(\001)1621 3798 y Fk(?)1621 3853 y(?)1621 3907 y(y)2133 3798 y(?)2133 3853 y(?)2133 3907 y(y)2193 3888 y Fq(\026)1610 4078 y Ft(X)92 b Fr(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)1883 4140 y Fq(d)2072 4078 y Ft(S)2133 4045 y Fp(2)2172 4078 y Ft(X)75 4315 y Fv(with)47 b Ft(d)i Fv(b)s(eing)e(the)i(diagonal)e(em)m(b)s(edding.)94 b(The)48 b(map)g Ft(\026)2245 4329 y Fp(\001)2318 4315 y Fv(:)36 b Ft(X)2461 4267 y Fp([2])2454 4344 y(\001)2596 4315 y Fr(!)55 b Ft(X)h Fv(equals)48 b(the)g Fs(P)3412 4282 y Fp(1)3452 4315 y Fv(-bundle)75 4423 y Fs(P)p Fv(\(\012)247 4437 y Fq(X)315 4423 y Fv(\))g Fr(!)h Ft(X)7 b Fv(.)82 b(W)-8 b(e)46 b(need)e(the)g(follo)m(wing)f(sligh)m(t)g(generalisation) g(of)i([)2684 4441 y SDict begin H.S end 2684 4441 a -18 x Fv(Huy06)2942 4361 y SDict begin H.R end 2942 4361 a 2942 4423 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Huy) cvn H.B /ANN pdfmark end 2942 4423 a Fv(,)i(Prop)s(osition)c (11.12])75 4531 y(for)30 b(a)h(blo)m(w-up)1665 4685 y Ft(E)1896 4633 y Fq(i)1783 4685 y Fr(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)2104 4662 y Fv(~)2078 4685 y Ft(X)1628 4852 y Fq(\031)1671 4756 y Fk(?)1671 4810 y(?)1671 4865 y(y)2089 4756 y(?)2089 4810 y(?)2089 4865 y(y)2149 4845 y Fq(p)1665 5036 y Ft(Y)65 b Fr(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)1892 5095 y Fq(j)2078 5036 y Ft(X)75 5230 y Fv(of)31 b(a)f(smo)s(oth)h(pro)5 b(jectiv)m(e)30 b(v)-5 b(ariet)m(y)31 b Ft(X)38 b Fv(along)30 b(a)h(smo)s(oth)f(sub)m(v)-5 b(ariet)m(y)30 b Ft(Y)51 b Fv(of)30 b(co)s(dimension)e Ft(c)p Fv(.)p eop end end %%Page: 8 8 TeXDict begin HPSdict begin 8 7 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 67 -8 a SDict begin H.S end 67 -8 a 67 -8 a SDict begin H.R end 67 -8 a 67 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.8) cvn /DEST pdfmark end 67 -8 a 75 66 a Fp(8)1508 b(ANDREAS)24 b(KR)n(UG)75 266 y SDict begin H.S end 75 266 a 75 266 a SDict begin 13 H.A end 75 266 a 75 266 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.2) cvn /DEST pdfmark end 75 266 a FF(Lemma)33 b(4.2.)42 b Fl(F)-7 b(or)34 b(every)e Fr(F)j(2)24 b Fv(Coh\()p Ft(Y)c Fv(\))33 b Fl(and)h(every)e Ft(k)d Fr(2)c Fs(Z)32 b Fl(ther)-5 b(e)34 b(is)e(an)h(isomorphism)1027 457 y Fr(H)1105 419 y Fq(k)1147 457 y Fv(\()p Ft(p)1228 419 y Fo(\003)1267 457 y Ft(j)1304 471 y Fo(\003)1344 457 y Fr(F)9 b Fv(\))1479 432 y Fr(\030)1479 461 y Fv(=)1575 457 y Ft(i)1606 471 y Fo(\003)1661 356 y Fk(\020)1715 457 y Ft(\031)1770 419 y Fo(\003)1810 457 y Fr(F)30 b(\012)19 b(^)2056 419 y Fo(\000)p Fq(k)2154 457 y Fv(\012)2220 471 y Fq(\031)2287 457 y Fr(\012)g(O)2449 471 y Fq(\031)2497 457 y Fv(\()p Fr(\000)p Ft(k)s Fv(\))2688 356 y Fk(\021)2773 457 y Ft(:)75 656 y Fl(Pr)-5 b(o)g(of.)43 b Fv(This)28 b(can)j(b)s(e)e(pro)m (v)m(en)i(lo)s(cally)-8 b(.)39 b(Hence,)32 b(w)m(e)e(ma)m(y)h(assume)f (that)h Ft(Y)45 b Fv(=)25 b Ft(Z)7 b Fv(\()p Ft(s)p Fv(\))30 b(is)f(the)i(zero)g(lo)s(cus)e(of)75 764 y(a)k(global)e(section)i(of)f (a)h(v)m(ector)h(bundle)c Fr(E)40 b Fv(of)33 b(rank)e Ft(c)p Fv(.)47 b(Th)m(us,)32 b(the)h(blo)m(w-up)d(diagram)i(can)h(b)s (e)e(enlarged)75 872 y(to)1396 1002 y Ft(E)1627 951 y Fq(i)1514 1002 y Fr(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)1835 979 y Fv(~)1809 1002 y Ft(X)2049 951 y Fq(\023)1937 1002 y Fr(\000)g(\000)-20 b(\000)f(\000)-42 b(!)45 b Fs(P)p Fv(\()p Fr(E)8 b Fv(\))1359 1170 y Fq(\031)1402 1074 y Fk(?)1402 1128 y(?)1402 1183 y(y)1820 1074 y(?)1820 1128 y(?)1820 1183 y(y)1880 1163 y Fq(p)2300 1074 y Fk(?)2300 1128 y(?)2300 1183 y(y)2360 1163 y Fq(g)1396 1354 y Ft(Y)65 b Fr(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)1623 1413 y Fq(j)1809 1354 y Ft(X)53 b Fr(\000)-43 b(\000)-20 b(\000)f(\000)-42 b(!)2032 1415 y Fp(id)2289 1354 y Ft(X)75 1543 y Fv(where)31 b Ft(\023)h Fv(is)f(a)h(closed)g(em)m(b)s(edding)e(of)i(co)s(dimension) e Ft(c)21 b Fr(\000)g Fv(1)32 b(suc)m(h)g(that)g(the)g(normal)f(bundle) e Ft(M)42 b Fv(has)32 b(the)75 1651 y(prop)s(ert)m(y)h Fr(^)511 1618 y Fq(k)554 1651 y Ft(M)652 1618 y Fo(_)642 1682 y(j)p Fq(E)752 1651 y Fv(=)f Fr(^)916 1618 y Fo(\000)p Fq(k)1013 1651 y Fv(\012)1079 1665 y Fq(\031)1148 1651 y Fr(\012)22 b(O)1313 1665 y Fq(\031)1361 1651 y Fv(\()p Fr(\000)p Ft(k)s Fv(\))34 b(\(see)h([)1797 1669 y SDict begin H.S end 1797 1669 a -18 x Fv(Huy06)2055 1589 y SDict begin H.R end 2055 1589 a 2055 1651 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Huy) cvn H.B /ANN pdfmark end 2055 1651 a Fv(,)g(p.)52 b(252]\).)h(The)34 b(outer)g(square)g(is)f (a)h(\015at)h(base)75 1770 y(c)m(hange.)42 b(It)30 b(follo)m(ws)g(that) 736 1934 y Ft(\023)768 1948 y Fo(\003)807 1934 y Ft(p)853 1896 y Fo(\003)893 1934 y Ft(j)930 1948 y Fo(\003)969 1934 y Fr(F)35 b(')25 b Ft(\023)1197 1948 y Fo(\003)1237 1934 y Ft(\023)1269 1896 y Fo(\003)1308 1934 y Ft(g)1354 1896 y Fo(\003)1394 1934 y Ft(j)1431 1948 y Fo(\003)1471 1934 y Fr(F)35 b(')25 b Ft(\023)1699 1948 y Fo(\003)1738 1934 y Ft(\023)1770 1896 y Fo(\003)1810 1934 y Ft(\023)1842 1948 y Fo(\003)1882 1934 y Ft(i)1913 1948 y Fo(\003)1952 1934 y Ft(\031)2007 1896 y Fo(\003)2047 1934 y Fr(F)34 b(')25 b Ft(\023)2274 1948 y Fo(\003)2314 1934 y Ft(i)2345 1948 y Fo(\003)2385 1934 y Fv(\()p Ft(\031)2475 1896 y Fo(\003)2515 1934 y Fr(F)k(\012)20 b Ft(i)2731 1896 y Fo(\003)2771 1934 y Ft(\023)2803 1896 y Fo(\003)2843 1934 y Ft(\023)2875 1948 y Fo(\003)2914 1934 y Fr(O)3006 1947 y Fp(~)2986 1964 y Fq(X)3054 1934 y Fv(\))75 1934 y SDict begin H.S end 75 1934 a 75 1934 a SDict begin 16.25 H.A end 75 1934 a 75 1934 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.5) cvn /DEST pdfmark end 75 1934 a Fv(\(5\))75 2097 y(where)42 b(the)g(last)g(isomorphism)d(is)i(giv)m(en)i(b)m(y)f (applying)e(the)i(pro)5 b(jection)42 b(form)m(ula)f(t)m(w)m(o)j(times.) 75 b(No)m(w)75 2205 y Fr(H)153 2172 y Fq(k)195 2205 y Fv(\()p Ft(\023)262 2172 y Fo(\003)302 2205 y Ft(\023)334 2219 y Fo(\003)374 2205 y Fr(O)466 2218 y Fp(~)446 2235 y Fq(X)514 2205 y Fv(\))574 2180 y Fr(\030)574 2209 y Fv(=)670 2205 y Fr(^)731 2172 y Fo(\000)p Fq(k)828 2205 y Ft(M)926 2172 y Fo(_)1008 2205 y Fv(is)29 b(lo)s(cally)g(free)i(for)f (ev)m(ery)h Ft(k)d Fr(2)d Fs(Z)p Fv(.)41 b(It)30 b(follo)m(ws)g(that) 124 2382 y Fr(H)202 2345 y Fq(k)244 2382 y Fv(\()p Ft(\031)334 2345 y Fo(\003)374 2382 y Fr(F)g(\012)20 b Ft(i)591 2345 y Fo(\003)630 2382 y Ft(\023)662 2345 y Fo(\003)702 2382 y Ft(\023)734 2396 y Fo(\003)774 2382 y Fr(O)866 2395 y Fp(~)846 2412 y Fq(X)913 2382 y Fv(\))974 2357 y Fr(\030)974 2386 y Fv(=)1070 2382 y Ft(\031)1125 2345 y Fo(\003)1164 2382 y Fr(F)30 b(\012)20 b Ft(i)1381 2345 y Fo(\003)1421 2382 y Fr(H)1499 2345 y Fq(k)1541 2382 y Fv(\()p Ft(\023)1608 2345 y Fo(\003)1648 2382 y Ft(\023)1680 2396 y Fo(\003)1720 2382 y Fr(O)1812 2395 y Fp(~)1792 2412 y Fq(X)1859 2382 y Fv(\))1920 2357 y Fr(\030)1920 2386 y Fv(=)2016 2382 y Ft(\031)2071 2345 y Fo(\003)2110 2382 y Fr(F)30 b(\012)20 b(^)2357 2345 y Fq(k)2399 2382 y Ft(M)2497 2345 y Fo(_)2487 2409 y(j)p Fq(E)2592 2357 y Fr(\030)2592 2386 y Fv(=)2688 2382 y Ft(\031)2743 2345 y Fo(\003)2782 2382 y Fr(F)30 b(\012)20 b(^)3029 2345 y Fo(\000)p Fq(k)3126 2382 y Fv(\012)3192 2396 y Fq(\031)3259 2382 y Fr(\012)g(O)3422 2396 y Fq(\031)3469 2382 y Fv(\()p Fr(\000)p Ft(k)s Fv(\))15 b Ft(:)75 2557 y Fv(By)31 b(\()253 2557 y SDict begin H.S end 253 2557 a Fv(5)298 2499 y SDict begin H.R end 298 2499 a 298 2557 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.5) cvn H.B /ANN pdfmark end 298 2557 a Fv(\))g(also)215 2727 y Ft(\023)247 2741 y Fo(\003)287 2727 y Fr(H)365 2690 y Fq(k)407 2727 y Fv(\()p Ft(p)488 2690 y Fo(\003)528 2727 y Ft(j)565 2741 y Fo(\003)605 2727 y Fr(F)9 b Fv(\))740 2702 y Fr(\030)740 2732 y Fv(=)836 2727 y Fr(H)914 2690 y Fq(k)956 2727 y Fv(\()p Ft(\023)1023 2741 y Fo(\003)1063 2727 y Ft(p)1109 2690 y Fo(\003)1148 2727 y Ft(j)1185 2741 y Fo(\003)1225 2727 y Fr(F)g Fv(\))1360 2702 y Fr(\030)1360 2732 y Fv(=)1456 2727 y Fr(H)1534 2690 y Fq(k)1591 2654 y Fk(\000)1633 2727 y Ft(\023)1665 2741 y Fo(\003)1705 2727 y Ft(i)1736 2741 y Fo(\003)1775 2727 y Fv(\()p Ft(\031)1865 2690 y Fo(\003)1905 2727 y Fr(F)30 b(\012)20 b Ft(i)2122 2690 y Fo(\003)2161 2727 y Ft(\023)2193 2690 y Fo(\003)2233 2727 y Ft(\023)2265 2741 y Fo(\003)2305 2727 y Fr(O)2397 2740 y Fp(~)2377 2757 y Fq(X)2444 2727 y Fv(\))2479 2654 y Fk(\001)2547 2702 y Fr(\030)2547 2732 y Fv(=)2643 2727 y Ft(\023)2675 2741 y Fo(\003)2714 2727 y Ft(i)2745 2741 y Fo(\003)2785 2727 y Fr(H)2863 2690 y Fq(k)2905 2727 y Fv(\()p Ft(\031)2995 2690 y Fo(\003)3035 2727 y Fr(F)30 b(\012)20 b Ft(i)3252 2690 y Fo(\003)3291 2727 y Ft(\023)3323 2690 y Fo(\003)3363 2727 y Ft(\023)3395 2741 y Fo(\003)3435 2727 y Fr(O)3527 2740 y Fp(~)3507 2757 y Fq(X)3574 2727 y Fv(\))75 2906 y(whic)m(h)29 b(pro)m(v)m(es)i(the)g(assertion)f(since) g Ft(\023)1408 2920 y Fo(\003)1457 2906 y Fv(:)46 b(Coh\()1751 2884 y(~)1725 2906 y Ft(X)7 b Fv(\))26 b Fr(!)f Fv(Coh\()p Fs(P)p Fv(\()p Fr(E)8 b Fv(\)\))32 b(is)d(fully)f(faithful.)621 b Fj(\003)75 3078 y SDict begin H.S end 75 3078 a 75 3078 a SDict begin 13 H.A end 75 3078 a 75 3078 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.3) cvn /DEST pdfmark end 75 3078 a FF(Remark)30 b(4.3.)39 b Fv(If)26 b Ft(X)34 b Fv(carries)26 b(an)h(action)g(b)m(y)f(a)h(\014nite)f(group)g Ft(G)h Fv(and)f Ft(Y)46 b Fv(is)26 b(in)m(v)-5 b(arian)m(t)26 b(under)f(this)g(action,)75 3188 y Ft(G)f Fv(also)h(acts)g(on)f(the)g (blo)m(w-up)1164 3165 y(~)1137 3188 y Ft(X)8 b Fv(.)38 b(The)24 b(bundle)e Ft(M)1844 3207 y Fo(j)p Fq(E)1947 3188 y Fv(of)i(the)h(pro)s(of)e(is)g(a)i(quotien)m(t)f(of)g(the)g (normal)f(bundle)75 3296 y Ft(N)148 3315 y Fq(E)t(=)p Fg(P)p Fp(\()p Fo(E)6 b Fp(\))426 3271 y Fr(\030)426 3300 y Fv(=)528 3296 y Ft(\031)583 3263 y Fo(\003)623 3296 y Ft(N)696 3315 y Fq(Y)i(=X)847 3296 y Fv(.)51 b(In)33 b(the)h(case)h(that)f(there)g(is)f(a)h(group)f(action)h(this)f(quotien) m(t)h(is)f Ft(G)p Fv(-equiv)-5 b(arian)m(t.)75 3408 y(Th)m(us,)28 b(the)h(form)m(ula)f(of)h(the)g(lemma)g(is)e(in)h(this)g(case)h(also)g (true)g(for)f Fr(F)35 b(2)25 b Fv(Coh)2781 3422 y Fq(G)2841 3408 y Fv(\()p Ft(X)7 b Fv(\))29 b(with)f(the)h(action)g(on)75 3516 y(the)e(righ)m(t)f(hand)f(side)h(induced)e(b)m(y)j(the)g (linearisation)d(of)i(the)h(w)m(edge)h(p)s(o)m(w)m(ers)e(of)h Ft(M)36 b Fv(resp)s(ectiv)m(ely)26 b Ft(N)3573 3534 y Fq(Y)8 b(=X)3725 3516 y Fv(.)75 3691 y SDict begin H.S end 75 3691 a 75 3691 a SDict begin 13 H.A end 75 3691 a 75 3691 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.4) cvn /DEST pdfmark end 75 3691 a FF(Prop)s(osition)45 b(4.4.)h Fl(L)-5 b(et)39 b Ft(X)47 b Fl(b)-5 b(e)39 b(a)h(smo)-5 b(oth)42 b(pr)-5 b(oje)g(ctive)40 b(surfac)-5 b(e)40 b(\(with)g(arbitr)-5 b(ary)42 b(c)-5 b(anonic)g(al)41 b(bund)5 b(le\).)75 3806 y(Then)33 b(ther)-5 b(e)33 b(is)g(an)g (isomorphism)j(of)d(functors)g Fv(\010)1829 3773 y Fo(\000)p Fp(1)1943 3806 y Fr(\016)21 b Ft(F)2105 3781 y Fr(\030)2105 3810 y Fv(=)2221 3783 y(~)2201 3806 y Ft(F)2259 3824 y Fo(O)2315 3832 y Fz(\026)2352 3850 y Fx(\001)2411 3824 y Fp(\()p Fo(\000)p Fp(2\))2560 3806 y Fv([1])p Fl(,)33 b(wher)-5 b(e)34 b Ft(F)k Fv(=)25 b Ft(\016)3205 3820 y Fo(\003)3265 3806 y Fr(\016)c Fv(triv)q Fl(.)75 4020 y(Pr)-5 b(o)g(of.)43 b Fv(The)34 b Fs(P)618 3987 y Fp(1)658 4020 y Fv(-bundles)f Ft(p)1073 4034 y Fp(\001)1146 4020 y Fv(:)f Ft(E)38 b Fr(!)33 b Ft(X)43 b Fv(and)34 b Ft(\026)1786 4034 y Fp(\001)1859 4020 y Fv(:)e Ft(X)1998 3972 y Fp([2])1991 4048 y(\001)2110 4020 y Fr(!)h Ft(X)42 b Fv(are)36 b(isomorphic)d(via)i Ft(q)13 b Fv(:)32 b Ft(I)3285 3987 y Fp(2)3324 4020 y Ft(X)41 b Fr(!)33 b Ft(X)3646 3987 y Fp([2])3725 4020 y Fv(.)75 4132 y(The)45 b(cen)m(tre)h(\001)f(of)g(the)h(blo)m(w-up)d Ft(p)10 b Fv(:)36 b Ft(I)1497 4099 y Fp(2)1536 4132 y Ft(X)58 b Fr(!)50 b Ft(X)1892 4099 y Fp(2)1977 4132 y Fv(has)45 b(co)s(dimension)e(2)i(in)f Ft(X)2980 4099 y Fp(2)3020 4132 y Fv(.)85 b(Th)m(us,)48 b Ft(p)3450 4099 y Fo(\003)3490 4132 y Ft(\016)3530 4146 y Fo(\003)3569 4132 y Fr(F)55 b Fv(is)75 4240 y(cohomologically)34 b(concen)m(trated)i (in)d(degree)h(0)h(and)e Fr(\000)p Fv(1.)52 b(Since)34 b Ft(p)2417 4207 y Fo(\003)2456 4240 y Ft(\016)2496 4254 y Fo(\003)2536 4240 y Fr(F)43 b Fv(is)33 b(concen)m(trated)j(on)e Ft(E)39 b Fv(where)75 4348 y Fm(S)150 4362 y Fp(2)215 4348 y Fv(is)24 b(acting)i(trivially)-8 b(,)24 b(one)h(can)h(tak)m(e)h (the)e(in)m(v)-5 b(arian)m(ts)24 b(ev)m(en)i(b)s(efore)f(applying)e (the)j(push-forw)m(ard)d(along)75 4457 y Ft(q)13 b Fv(:)31 b Ft(I)232 4424 y Fp(2)271 4457 y Ft(X)j Fr(!)26 b Ft(X)579 4424 y Fp([2])658 4457 y Fv(.)42 b(The)30 b(group)h Fm(S)1246 4471 y Fp(2)1316 4457 y Fv(acts)h(on)f Fr(^)1692 4424 y Fp(0)1731 4457 y Ft(N)1804 4477 y Fp(\001)p Fq(=X)1961 4458 y Fx(2)2031 4457 y Fv(trivially)d(and)i(on)h Ft(N)2752 4477 y Fp(\001)p Fq(=X)2909 4458 y Fx(2)2979 4457 y Fv(alternating.)42 b(By)31 b(the)75 4577 y(previous)26 b(lemmas)h(and)g(the)h(remark)f(it) g(follo)m(ws)g(that)h(there)g(is)e(a)i(natural)f(isomorphism)e(\010) 3276 4544 y Fo(\000)p Fp(1)3384 4577 y Fr(\016)15 b Ft(F)e Fv(\()p Ft(A)p Fv(\))26 b Fr(')95 4669 y Fv(~)75 4692 y Ft(F)133 4711 y Fo(O)189 4719 y Fz(\026)226 4736 y Fx(\001)285 4711 y Fp(\()p Fo(\000)p Fp(2\))434 4692 y Fv(\()p Ft(A)p Fv(\)[1])35 b(for)e Ft(A)d Fr(2)f Fv(Coh\()p Ft(X)7 b Fv(\).)49 b(Using)33 b(the)g(sp)s(ectral)f(sequence)i(asso)s (ciated)f(to)h(the)f(pull-bac)m(k)f Ft(q)3711 4659 y Fo(\003)75 4807 y Fv(yields)d(the)h(isomorphism)e(of)i(functors.)2210 b Fj(\003)75 4981 y SDict begin H.S end 75 4981 a 75 4981 a SDict begin 13 H.A end 75 4981 a 75 4981 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.5) cvn /DEST pdfmark end 75 4981 a FF(Remark)33 b(4.5.)41 b Fv(The)29 b(prop)s(osition)e (sa)m(ys)i(in)f(particular)g(that)2268 4958 y(~)2247 4981 y Ft(F)2305 4999 y Fo(O)2361 5007 y Fz(\026)2398 5024 y Fx(\001)2457 4999 y Fp(\()p Fo(\000)p Fp(2\))2635 4981 y Fv(is)h(also)g(a)g(spherical)f(functor)g(in)75 5096 y(the)j(case)h(that)g Ft(!)680 5110 y Fq(X)778 5096 y Fv(is)e(not)h(trivial.)41 b(One)31 b(can)g(also)g(pro)m(v)m(e)h(this) e(directly)f(and)i(for)g(general)g Ft(L)f Fv(instead)h(of)75 5204 y Fr(O)147 5218 y Fq(\026)189 5229 y Fx(\001)248 5204 y Fv(\()p Fr(\000)p Fv(2\).)p eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 67 -8 a SDict begin H.S end 67 -8 a 67 -8 a SDict begin H.R end 67 -8 a 67 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.9) cvn /DEST pdfmark end 67 -8 a 329 66 a Fp(NEW)30 b(DERIVED)h(A)n(UTOEQUIV)-8 b(ALENCES)30 b(OF)h(HILBER)-6 b(T)31 b(SCHEMES)f(AND)g(KUMMER)g(V)-8 b(ARIETIES)219 b(9)75 269 y SDict begin H.S end 75 269 a 75 269 a SDict begin 13 H.A end 75 269 a 75 269 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.6) cvn /DEST pdfmark end 75 269 a FF(Remark)44 b(4.6.)i Fv(Let)882 246 y(^)861 269 y Ft(F)52 b Fv(:=)39 b Ft(M)1194 283 y Fn(a)1257 269 y Fr(\016)26 b Ft(F)d Fv(:)49 b(D)1552 233 y Fq(b)1587 269 y Fv(\()p Ft(X)7 b Fv(\))40 b Fr(!)e Fv(D)1978 233 y Fq(b)2012 269 y Fv(\()p Ft(X)2129 236 y Fp(2)2169 269 y Fv(\))h(where)f Ft(M)2602 283 y Fn(a)2678 269 y Fv(is)g(the)g(tensor) h(pro)s(duct)e(with)75 377 y(the)31 b(alternating)f(represen)m(tation)h Fm(a)p Fv(.)43 b(It)31 b(is)f(again)h(a)g(spherical)e(functor)i(b)m(y)g (lemma)3032 377 y SDict begin H.S end 3032 377 a Fv(2.3)3148 318 y SDict begin H.R end 3148 318 a 3148 377 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.3) cvn H.B /ANN pdfmark end 3148 377 a Fv(.)42 b(F)-8 b(ollo)m(wing)31 b(the)75 489 y(pro)s(of)f(of)g(prop)s(osition)894 489 y SDict begin H.S end 894 489 a Fv(4.4)1010 430 y SDict begin H.R end 1010 430 a 1010 489 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.4) cvn H.B /ANN pdfmark end 1010 489 a 31 w Fv(one)g(gets)i(that)f(\010)1658 456 y Fo(\000)p Fp(1)1772 489 y Fr(\016)1858 466 y Fv(^)1838 489 y Ft(F)1934 464 y Fr(\030)1934 493 y Fv(=)2050 466 y(~)2030 489 y Ft(F)2088 507 y Fo(O)2144 515 y Fz(\026)2181 532 y Fx(\001)2240 507 y Fp(\()p Fo(\000)p Fp(1\))2389 489 y Fv(.)75 702 y SDict begin H.S end 75 702 a 75 702 a SDict begin 13 H.A end 75 702 a 75 702 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.7) cvn /DEST pdfmark end 75 702 a FF(Remark)k(4.7.)43 b Fv(Since)30 b Ft(X)998 654 y Fp([2])991 731 y(\001)1109 702 y Fv(equals)g(the)i Fs(P)1614 669 y Fp(1)1654 702 y Fv(-bundle)d Fs(P)p Fv(\(\012)2155 716 y Fq(X)2222 702 y Fv(\))j(o)m(v)m(er)h Ft(X)7 b Fv(,)32 b(its)f(canonical)g(bundle) e(is)h(giv)m(en)75 815 y(b)m(y)36 b Ft(!)264 856 y Fq(X)327 821 y Fx([2])322 879 y(\001)435 815 y Fv(=)e Ft(\026)595 782 y Fo(\003)595 842 y Fp(\001)658 815 y Ft(!)718 782 y Fp(2)715 842 y Fq(X)805 815 y Fr(\012)24 b(O)972 829 y Fq(\026)1014 840 y Fx(\001)1073 815 y Fv(\()p Fr(\000)p Fv(2\))36 b(\(see)h(e.g.)g([)1680 815 y SDict begin H.S end 1680 815 a Fv(F)-8 b(ul98)1899 752 y SDict begin H.R end 1899 752 a 1899 815 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Ful) cvn H.B /ANN pdfmark end 1899 815 a Fv(,)37 b(B)g(5.8]\).)58 b(The)36 b(Hilb)s(ert-Cho)m(w)e(morphism)f Ft(\026)j Fv(is)f(a)75 951 y(crepan)m(t)c(resolution,)e(i.e.)i Ft(!)1050 977 y Fq(X)1113 957 y Fx([2])1212 926 y Fr(\030)1212 956 y Fv(=)1307 951 y Ft(\026)1362 918 y Fo(\003)1402 951 y Ft(!)1459 971 y Fq(S)1506 952 y Fx(2)1540 971 y Fq(X)1607 951 y Fv(.)41 b(Th)m(us,)1241 1120 y Ft(!)1298 1161 y Fq(X)1361 1142 y Fx([2])1431 1161 y Fo(j)p Fq(X)1514 1126 y Fx([2])1509 1185 y(\001)1613 1095 y Fr(\030)1613 1124 y Fv(=)1708 1120 y Ft(\026)1763 1083 y Fo(\003)1763 1143 y Fp(\001)1826 1120 y Fv(\()p Ft(!)1918 1140 y Fq(S)1965 1121 y Fx(2)1999 1140 y Fq(X)5 b Fo(j)p Fp(\001)2145 1120 y Fv(\))2206 1095 y Fr(\030)2206 1124 y Fv(=)2302 1120 y Ft(\026)2357 1083 y Fo(\003)2357 1143 y Fp(\001)2419 1120 y Ft(!)2479 1083 y Fp(2)2476 1143 y Fq(X)2559 1120 y Ft(:)75 1345 y Fv(It)30 b(follo)m(ws)f(b)m(y)h(adjunction)f(form)m (ula)g(that)i Fr(O)1651 1386 y Fq(X)1714 1351 y Fx([)p Fz(n)p Fx(])1709 1410 y(\001)1796 1345 y Fv(\()p Ft(X)1913 1297 y Fp([)p Fq(n)p Fp(])1906 1374 y(\001)2000 1345 y Fv(\))26 b(=)f Fr(O)2229 1359 y Fq(\026)2271 1370 y Fx(\001)2330 1345 y Fv(\()p Fr(\000)p Fv(2\).)41 b(The)30 b(line)e(bundle)g Fr(O)3310 1371 y Fq(X)3373 1351 y Fx([2])3447 1345 y Fv(\()p Fr(\000)p Ft(X)3635 1297 y Fp([2])3628 1374 y(\001)3715 1345 y Fv(\))75 1516 y(has)37 b(a)g(square)g(ro)s(ot)g Ft(D)i Fv(:=)d Fr(O)s Fv(\()p Fr(\000)p Ft(X)1329 1468 y Fp([2])1322 1544 y(\001)1408 1516 y Ft(=)p Fv(2\))i Fr(2)d Fv(Pic)15 b Ft(X)1892 1483 y Fp([2])2008 1516 y Fv(\(see)38 b([)2222 1516 y SDict begin H.S end 2222 1516 a Fv(Leh99)2461 1452 y SDict begin H.R end 2461 1452 a 2461 1516 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Leh) cvn H.B /ANN pdfmark end 2461 1516 a Fv(,)h(lemma)d(3.8]\).)62 b(Its)37 b(restriction)f(to)75 1650 y Ft(X)157 1602 y Fp([2])150 1679 y(\001)275 1650 y Fv(is)i(of)h(the)g(form)f Ft(D)950 1692 y Fo(j)p Fq(X)1033 1656 y Fx([2])1028 1715 y(\001)1146 1650 y Fv(=)i Fr(O)1329 1664 y Fq(\026)1371 1675 y Fx(\001)1429 1650 y Fv(\(1\).)68 b(Using)38 b(this,)i(w)m(e)f (can)g(rewrite)f(for)h(a)g(general)g(line)e(bundle)75 1821 y Ft(L)25 b Fv(=)g Ft(\026)313 1788 y Fo(\003)313 1848 y Fp(\001)376 1821 y Ft(K)f Fr(\012)17 b(O)637 1835 y Fq(\026)679 1846 y Fx(\001)738 1821 y Fv(\()p Ft(i)p Fv(\))27 b Fr(2)d Fv(Pic)15 b Ft(X)1176 1773 y Fp([2])1169 1849 y(\001)1284 1821 y Fv(the)29 b(spherical)f(functor)2153 1798 y(~)2132 1821 y Ft(F)2190 1835 y Fq(L)2272 1821 y Fv(as)2403 1798 y(~)2382 1821 y Ft(F)2440 1835 y Fq(L)2518 1821 y Fv(=)d Ft(M)2712 1782 y Fq(i)p Fp(+2)2702 1849 y Fq(D)2848 1821 y Fr(\016)2931 1798 y Fv(~)2911 1821 y Ft(F)2969 1835 y Fo(O)3025 1849 y Fz(\026)3062 1866 y Fx(\001)3117 1849 y(\()p Fy(\000)p Fx(2\))3269 1821 y Fr(\016)18 b Ft(M)3420 1835 y Fq(K)3517 1821 y Fv(where)75 1938 y Ft(M)163 1952 y Fq(K)267 1938 y Fv(denotes)36 b(the)g(auto)s(equiv)-5 b(alence)35 b(giv)m(en)g(b)m(y)h(tensor)f(pro)s (duct)g(with)f Ft(K)7 b Fv(.)56 b(The)35 b(analogue)h(of)f(lemma)75 2061 y SDict begin H.S end 75 2061 a Fv(2.3)191 2003 y SDict begin H.R end 191 2003 a 191 2061 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.3) cvn H.B /ANN pdfmark end 191 2061 a 27 w Fv(for)28 b(spherical)d(functors)i(th)m(us)g(yields)e Ft(t)1562 2075 y Fq(L)1640 2061 y Fv(=)g Ft(M)1834 2023 y Fq(i)p Fp(+2)1824 2090 y Fq(D)1966 2061 y Fr(\016)14 b Ft(t)2058 2080 y Fo(O)2114 2088 y Fz(\026)2151 2105 y Fx(\001)2210 2080 y Fp(\()p Fo(\000)p Fp(2\))2373 2061 y Fr(\016)g Ft(M)2530 2013 y Fo(\000)p Fp(\()p Fq(i)p Fp(+2\))2520 2090 y Fq(D)2787 2061 y Fv(where)27 b Ft(t)3080 2075 y Fq(L)3159 2061 y Fv(is)f(the)i(spherical)75 2197 y(t)m(wist)i(asso)s(ciated)h(to)865 2174 y(~)844 2197 y Ft(F)902 2211 y Fq(L)955 2197 y Fv(.)75 2368 y SDict begin H.S end 75 2368 a 75 2368 a SDict begin 13 H.A end 75 2368 a 75 2368 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.8) cvn /DEST pdfmark end 75 2368 a FF(Remark)38 b(4.8.)44 b Fv(F)-8 b(or)34 b(general)g Ft(n)c Fr(\025)g Fv(2)k(ev)m(ery)h(ob)5 b(ject)34 b(in)e(the)i(image)g(of)g(\010)2660 2335 y Fo(\000)p Fp(1)2776 2368 y Fr(\016)23 b Ft(F)46 b Fv(still)32 b(is)g(supp)s(orted)g(on)75 2492 y Ft(X)157 2444 y Fp([)p Fq(n)p Fp(])150 2521 y(\001)269 2492 y Fv(=)25 b Ft(\026)420 2459 y Fo(\000)p Fp(1)514 2492 y Fv(\(\001\).)75 2644 y SDict begin H.S end 75 2644 a 75 2644 a SDict begin 13 H.A end 75 2644 a 75 2644 a SDict begin [/View [/XYZ H.V]/Dest (section.5) cvn /DEST pdfmark end 75 2644 a 885 2723 a Fv(5.)46 b Fu(Comp)-6 b(arison)33 b(with)i(other)e(a)n(utoequiv)-8 b(alences)175 2885 y Fv(In)44 b(the)h(follo)m(wing)e(w)m(e)j(will)c(denote)j(the)g Fs(P)1756 2852 y Fq(n)1804 2885 y Fv(-t)m(wist)g(asso)s(ciated)g(to)h Ft(F)58 b Fv(resp)s(ectiv)m(ely)44 b(\010)3339 2852 y Fo(\000)p Fp(1)3463 2885 y Fr(\016)30 b Ft(F)58 b Fv(b)m(y)75 2994 y Ft(b)30 b Fr(2)g Fv(Aut\(D)494 2958 y Fq(b)494 3018 y Fn(S)549 3026 y Fz(n)595 2994 y Fv(\()p Ft(X)712 2961 y Fq(n)760 2994 y Fv(\)\))861 2969 y Fr(\030)861 2998 y Fv(=)962 2994 y(Aut\(D)1220 2958 y Fq(b)1255 2994 y Fv(\()p Ft(X)1372 2961 y Fp([)p Fq(n)p Fp(])1459 2994 y Fv(\)\).)50 b(In)32 b(the)i(case)g(that)g Ft(n)29 b Fv(=)h(2)k(the)f(functor)g Ft(F)46 b Fv(is)33 b(spherical)e(\(see)75 3120 y(example)431 3120 y SDict begin H.S end 431 3120 a Fv(2.2)547 3061 y SDict begin H.R end 547 3061 a 547 3120 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.2) cvn H.B /ANN pdfmark end 547 3120 a 31 w Fv(\(ii\)\).)40 b(W)-8 b(e)31 b(denote)g(the)g(asso)s(ciated)g(spherical)d(t)m(wist)i (b)m(y)2570 3042 y Fr(p)p 2645 3042 40 4 v 2645 3120 a Ft(b)p Fv(.)75 3291 y SDict begin H.S end 75 3291 a 75 3291 a SDict begin 13 H.A end 75 3291 a 75 3291 a SDict begin [/View [/XYZ H.V]/Dest (theorem.5.1) cvn /DEST pdfmark end 75 3291 a FF(Prop)s(osition)40 b(5.1.)j Fl(The)36 b(automorphism)i Ft(b)e Fl(is)f(not)h(c)-5 b(ontaine)g(d)37 b(in)e(the)h(gr)-5 b(oup)37 b(of)e(standar)-5 b(d)38 b(automor-)75 3399 y(phisms)668 3546 y Fv(Aut\(D)927 3509 y Fq(b)961 3546 y Fv(\()p Ft(X)1078 3509 y Fp([)p Fq(n)p Fp(])1165 3546 y Fv(\)\))26 b Fr(\033)f Fv(D)m(Aut)1578 3560 y Fq(st)1640 3546 y Fv(\()p Ft(X)1757 3509 y Fp([)p Fq(n)p Fp(])1844 3546 y Fv(\))1905 3521 y Fr(\030)1905 3550 y Fv(=)2001 3546 y Fs(Z)20 b Fr(\002)2177 3445 y Fk(\020)2231 3546 y Fv(Aut\()p Ft(X)2502 3509 y Fp([)p Fq(n)p Fp(])2589 3546 y Fv(\))h Ff(n)f Fv(Pic)o(\()p Ft(X)2980 3509 y Fp([)p Fq(n)p Fp(])3067 3546 y Fv(\))3102 3445 y Fk(\021)75 3712 y Fl(gener)-5 b(ate)g(d)41 b(by)f(shifts,)j (push-forwar)-5 b(ds)43 b(along)e(automorphisms)j(and)c(taking)h (tensor)g(pr)-5 b(o)g(ducts)42 b(by)e(line)75 3823 y(bund)5 b(les.)42 b(The)33 b(same)g(holds)h(in)f(the)g(c)-5 b(ase)33 b Ft(n)25 b Fv(=)g(2)33 b Fl(for)1947 3746 y Fr(p)p 2023 3746 40 4 v 77 x Ft(b)p Fl(.)75 4010 y(Pr)-5 b(o)g(of.)43 b Fv(Let)36 b([)p Ft(\030)t Fv(])f Fr(2)e Ft(X)829 3977 y Fp([)p Fq(n)p Fp(])940 4010 y Fr(n)24 b Ft(X)1091 3962 y Fp([)p Fq(n)p Fp(])1084 4039 y(\001)1177 4010 y Fv(,)38 b(i.e.)d Fr(j)15 b Fv(supp)f Ft(\030)t Fr(j)35 b(\025)e Fv(2.)57 b(Then)35 b(b)m(y)h(remark)2660 4010 y SDict begin H.S end 2660 4010 a Fv(4.8)2776 3952 y SDict begin H.R end 2776 3952 a 2776 4010 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.8) cvn H.B /ANN pdfmark end 2776 4010 a 36 w Fv(and)f(prop)s(osition)3477 4010 y SDict begin H.S end 3477 4010 a Fv(2.1)3593 3952 y SDict begin H.R end 3593 3952 a 3593 4010 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 3593 4010 a 36 w Fv(\(i\),)75 4130 y(w)m(e)g(ha)m(v)m(e)g Ft(b)p Fv(\()p Fs(C)p Fv(\([)p Ft(\030)t Fv(]\)\))f(=)e Fs(C)p Fv(\([)p Ft(\030)t Fv(]\).)53 b(Let)35 b Ft(g)h Fv(=)31 b([)p Ft(`)p Fv(])23 b Fr(\016)h Ft(')1815 4144 y Fo(\003)1877 4130 y Fr(\016)g Ft(M)2034 4144 y Fq(L)2118 4130 y Fr(2)31 b Fv(D)m(Aut)2431 4144 y Fq(st)2494 4130 y Fv(\()p Ft(X)2611 4097 y Fp([)p Fq(n)p Fp(])2697 4130 y Fv(\))k(where)f Ft(M)3122 4144 y Fq(L)3209 4130 y Fv(is)f(the)h (functor)75 4244 y Ft(E)e Fr(7!)27 b Ft(E)g Fr(\012)20 b Ft(L)31 b Fv(for)h(an)f Ft(L)c Fr(2)g Fv(Pic)14 b Ft(X)1239 4211 y Fp([)p Fq(n)p Fp(])1326 4244 y Fv(.)44 b(Then)30 b Ft(g)s Fv(\()p Fs(C)p Fv(\([)p Ft(\030)t Fv(]\)\))g(=)d Fs(C)p Fv(\()p Ft(')p Fv(\([)p Ft(\030)t Fv(]\)\)[)p Ft(`)p Fv(].)46 b(Th)m(us,)31 b(the)h(assumption)e Ft(b)d Fv(=)f Ft(g)75 4373 y Fv(implies)f Ft(`)g Fv(=)g(0)k(and)e(also)i Ft(')c Fv(=)g(id)o(,)k(since)e Ft(X)1581 4340 y Fp([)p Fq(n)p Fp(])1684 4373 y Fr(n)16 b Ft(X)1827 4325 y Fp([)p Fq(n)p Fp(])1820 4401 y(\001)1941 4373 y Fv(is)28 b(op)s(en)f(in)g Ft(X)2434 4340 y Fp([)p Fq(n)p Fp(])2520 4373 y Fv(.)40 b(Th)m(us,)28 b(the)g(only)g(p)s(ossibilit)m(y)c(left)75 4492 y(for)k Ft(b)d Fr(2)g Fv(D)m(Aut)583 4506 y Fq(st)646 4492 y Fv(\()p Ft(X)763 4459 y Fp([)p Fq(n)p Fp(])850 4492 y Fv(\))j(is)g Ft(b)d Fv(=)g Ft(M)1251 4506 y Fq(L)1332 4492 y Fv(for)j(some)h(line)d(bundle)h Ft(L)h Fv(whic)m(h)f(can)i(not)f (hold)f(b)m(y)i(prop)s(osition)3634 4492 y SDict begin H.S end 3634 4492 a Fv(2.1)3750 4434 y SDict begin H.R end 3750 4434 a 3750 4492 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 3750 4492 a 75 4608 a Fv(\(ii\).)40 b(The)30 b(pro)s(of)f(that)883 4531 y Fr(p)p 959 4531 40 4 v 77 x Ft(b)35 b(=)-55 b Fr(2)25 b Fv(D)m(Aut)1330 4622 y Fq(st)1392 4608 y Fv(\()p Ft(X)1509 4575 y Fp([)p Fq(n)p Fp(])1596 4608 y Fv(\))31 b(is)e(the)i(same.)1547 b Fj(\003)175 4788 y Fv(In)31 b([)315 4788 y SDict begin H.S end 315 4788 a Fv(Plo07)538 4725 y SDict begin H.R end 538 4725 a 538 4788 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Plo) cvn H.B /ANN pdfmark end 538 4788 a Fv(])h(Plo)s(og)g(ga)m(v)m(e)i(a)e(general)f (construction)g(whic)m(h)g(asso)s(ciates)h(to)h(deriv)m(ed)d(auto)s (equiv)-5 b(alences)75 4902 y(of)33 b(the)f(surface)h Ft(X)40 b Fv(deriv)m(ed)31 b(auto)s(equiv)-5 b(alences)32 b(of)h(the)g(Hilb)s(ert)d(sc)m(heme)k Ft(X)2756 4869 y Fp([)p Fq(n)p Fp(])2842 4902 y Fv(.)47 b(Let)33 b(\011)c Fr(2)f Fv(Aut\(D)3527 4866 y Fq(b)3561 4902 y Fv(\()p Ft(X)7 b Fv(\)\))75 5015 y(with)24 b(F)-8 b(ourier{Muk)j(ai)26 b(k)m(ernel)f Fr(P)33 b(2)25 b Fv(D)1395 4979 y Fq(b)1429 5015 y Fv(\()p Ft(X)19 b Fr(\002)11 b Ft(X)c Fv(\).)40 b(The)25 b(ob)5 b(ject)27 b Fr(P)2345 4982 y Fh(\002)p Fq(n)2469 5015 y Fr(2)e Fv(D)2624 4979 y Fq(b)2659 5015 y Fv(\()p Ft(X)2776 4982 y Fq(n)2834 5015 y Fr(\002)11 b Ft(X)2998 4982 y Fq(n)3046 5015 y Fv(\))26 b(carries)f(a)h(natural)75 5122 y Fm(S)150 5136 y Fq(n)197 5122 y Fv(-linearisation)k(giv)m(en)h (b)m(y)g(p)s(erm)m(utation)g(of)h(the)f(b)s(o)m(x)h(factors.)45 b(Th)m(us,)31 b(it)g(induces)e(a)j Fm(S)3230 5136 y Fq(n)3278 5122 y Fv(-equiv)-5 b(arian)m(t)75 5230 y(deriv)m(ed)29 b(auto)s(equiv)-5 b(alence)31 b Ft(\013)p Fv(\(\011\))26 b(:=)f(FM)1542 5255 y Fo(P)1597 5236 y Fe(\002)p Fz(n)1718 5230 y Fv(of)30 b Ft(X)1903 5197 y Fq(n)1951 5230 y Fv(.)40 b(This)29 b(giv)m(es)i(the)f(follo)m(wing.)p eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 67 -8 a SDict begin H.S end 67 -8 a 67 -8 a SDict begin H.R end 67 -8 a 67 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.10) cvn /DEST pdfmark end 67 -8 a 75 66 a Fp(10)1473 b(ANDREAS)24 b(KR)n(UG)75 266 y SDict begin H.S end 75 266 a 75 266 a SDict begin 13 H.A end 75 266 a 75 266 a SDict begin [/View [/XYZ H.V]/Dest (theorem.5.2) cvn /DEST pdfmark end 75 266 a FF(Theorem)34 b(5.2)h Fv(\([)747 266 y SDict begin H.S end 747 266 a Fv(Plo07)971 203 y SDict begin H.R end 971 203 a 971 266 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Plo) cvn H.B /ANN pdfmark end 971 266 a Fv(]\))p FF(.)43 b Fl(The)32 b(ab)-5 b(ove)34 b(c)-5 b(onstruction)34 b(gives)e(an)h(inje)-5 b(ctive)32 b(gr)-5 b(oup)34 b(homomorphism)869 443 y Ft(\013)10 b Fv(:)46 b(Aut\(D)1267 405 y Fq(b)1301 443 y Fv(\()p Ft(X)7 b Fv(\)\))27 b Fr(!)e Fv(Aut\(D)1890 405 y Fq(b)1890 466 y Fn(S)1945 474 y Fz(n)1991 443 y Fv(\()p Ft(X)2108 405 y Fq(n)2156 443 y Fv(\)\))2252 418 y Fr(\030)2252 447 y Fv(=)2348 443 y(Aut\(D)2607 405 y Fq(b)2641 443 y Fv(\()p Ft(X)2758 405 y Fp([)p Fq(n)p Fp(])2845 443 y Fv(\)\))15 b Ft(:)75 616 y SDict begin H.S end 75 616 a 75 616 a SDict begin 13 H.A end 75 616 a 75 616 a SDict begin [/View [/XYZ H.V]/Dest (theorem.5.3) cvn /DEST pdfmark end 75 616 a FF(Remark)32 b(5.3.)41 b Fv(F)-8 b(or)30 b(ev)m(ery)f Ft(')d Fr(2)f Fv(Aut\()p Ft(X)7 b Fv(\))30 b(w)m(e)f(ha)m(v)m(e)h Ft(\013)p Fv(\()p Ft(')2072 630 y Fo(\003)2112 616 y Fv(\))c(=)f(\()p Ft(')2363 584 y Fq(n)2411 616 y Fv(\))2446 630 y Fo(\003)2514 616 y Fv(where)j Ft(')2834 584 y Fq(n)2911 616 y Fv(is)f(the)i Fm(S)3230 630 y Fq(n)3278 616 y Fv(-equiv)-5 b(arian)m(t)75 724 y(automorphism)35 b(of)h Ft(X)874 691 y Fq(n)958 724 y Fv(giv)m(en)h(b)m(y)f Ft(')p Fv(\()p Ft(x)1480 738 y Fp(1)1520 724 y Ft(;)15 b(:)g(:)g(:)i(;)e(x)1774 738 y Fq(n)1821 724 y Fv(\))36 b(=)f(\()p Ft(')p Fv(\()p Ft(x)2179 738 y Fp(1)2219 724 y Fv(\))p Ft(;)15 b(:)g(:)g(:)i(;)e(')p Fv(\()p Ft(x)2602 738 y Fq(n)2650 724 y Fv(\)\).)60 b(F)-8 b(urthermore,)38 b Ft(')f Fv(acts)g(on)75 838 y Ft(X)157 805 y Fp([)p Fq(n)p Fp(])274 838 y Fv(b)m(y)30 b(the)h(morphism)d Ft(')1041 805 y Fp([)p Fq(n)p Fp(])1127 838 y Fv(,)j(whic)m(h)e(is)g (giv)m(en)i(b)m(y)f Ft(')1957 805 y Fp([)p Fq(n)p Fp(])2043 838 y Fv(\([)p Ft(\030)t Fv(]\))d(=)e([)p Ft(')p Fv(\()p Ft(\030)t Fv(\)],)32 b(and)e(on)g Ft(X)2995 805 y Fq(n)3073 838 y Fv(b)m(y)g(the)g(morphism)75 952 y Ft(')134 919 y Fq(n)181 952 y Fv(.)40 b(The)25 b(isosp)s(ectral)g(Hilb)s(ert)f(sc)m (heme)j Ft(I)1535 919 y Fq(n)1582 952 y Ft(X)33 b Fr(\032)25 b Ft(X)1868 919 y Fp([)p Fq(n)p Fp(])1966 952 y Fr(\002)12 b Ft(X)2131 919 y Fq(n)2204 952 y Fv(is)25 b(in)m(v)-5 b(arian)m(t)25 b(under)g(the)h(induced)e(action)j(of)75 1061 y(Aut\()p Ft(X)7 b Fv(\))27 b(on)e Ft(X)611 1028 y Fp([)p Fq(n)p Fp(])708 1061 y Fr(\002)11 b Ft(X)872 1028 y Fq(n)919 1061 y Fv(.)39 b(Th)m(us,)26 b(the)g (Bridgeland{King{Reid{Haiman)c(equiv)-5 b(alence)25 b(\010)g(=)g(FM) 3472 1075 y Fo(O)3528 1088 y Fz(I)3559 1074 y(n)3601 1088 y(X)3689 1061 y Fv(is)75 1179 y(Aut\()p Ft(X)7 b Fv(\)-equiv)-5 b(arian)m(t)30 b(in)d(the)i(sense)g(that)h(\010)16 b Fr(\016)i Fv(\()p Ft(')1810 1146 y Fp([)p Fq(n)p Fp(])1897 1179 y Fv(\))1932 1193 y Fo(\003)1997 1179 y Fr(')25 b Fv(\()p Ft(')2187 1146 y Fq(n)2235 1179 y Fv(\))2270 1193 y Fo(\003)2327 1179 y Fr(\016)18 b Fv(\010.)40 b(Hence,)30 b Ft(\013)p Fv(\()p Ft(')2967 1193 y Fo(\003)3008 1179 y Fv(\))25 b Fr(2)g Fv(Aut\(D)3413 1143 y Fq(b)3413 1204 y Fn(S)3468 1212 y Fz(n)3515 1179 y Fv(\()p Ft(X)3632 1146 y Fq(n)3679 1179 y Fv(\)\))75 1318 y(corresp)s(onds)37 b(to)i Ft(')760 1270 y Fp([)p Fq(n)p Fp(])760 1332 y Fo(\003)885 1318 y Fr(2)g Fv(Aut\(D)1243 1282 y Fq(b)1278 1318 y Fv(\()p Ft(X)1395 1285 y Fp([)p Fq(n)p Fp(])1482 1318 y Fv(\)\).)65 b(F)-8 b(or)39 b Ft(L)g Fr(2)f Fv(Pic)14 b Ft(X)46 b Fv(w)m(e)39 b(ha)m(v)m(e)h Ft(\013)p Fv(\()p Ft(M)2817 1332 y Fq(L)2870 1318 y Fv(\))f(=)f Ft(M)3141 1343 y Fq(L)3189 1324 y Fe(\002)p Fz(n)3319 1318 y Fv(where)g Ft(L)3652 1285 y Fh(\002)p Fq(n)75 1426 y Fv(is)e(considered)g(as)h(a)g Fm(S)900 1440 y Fq(n)947 1426 y Fv(-equiv)-5 b(arian)m(t)37 b(line)e(bundle)g(with)g(the)i(natural)f(linearisation.)58 b(Under)36 b(\010,)j(the)75 1535 y(automorphism)34 b Ft(M)770 1559 y Fq(L)818 1540 y Fe(\002)p Fz(n)944 1535 y Fv(corresp)s(onds)g(to)i Ft(M)1652 1549 y Fo(D)1707 1560 y Fz(L)1790 1535 y Fr(2)d Fv(Aut\(D)2143 1499 y Fq(b)2177 1535 y Fv(\()p Ft(X)2294 1502 y Fp([)p Fq(n)p Fp(])2381 1535 y Fv(\)\))j(where)f Fr(D)2825 1549 y Fq(L)2911 1535 y Fr(2)d Fv(Pic)15 b Ft(X)3229 1502 y Fp([)p Fq(n)p Fp(])3351 1535 y Fv(is)34 b(the)i(line)75 1647 y(bundle)28 b Fr(D)443 1661 y Fq(L)521 1647 y Fv(:=)d Ft(\026)697 1614 y Fo(\003)736 1647 y Fv(\(\()p Ft(L)868 1614 y Fh(\002)p Fq(n)967 1647 y Fv(\))1002 1614 y Fn(S)1057 1622 y Fz(n)1104 1647 y Fv(\))31 b(\(see)g(e.g.)h([)1544 1647 y SDict begin H.S end 1544 1647 a Fv(Kru12)1792 1585 y SDict begin H.R end 1792 1585 a 1792 1647 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Kru2) cvn H.B /ANN pdfmark end 1792 1647 a Fv(,)e(lemma)g(9.2]\).)75 1736 y SDict begin H.S end 75 1736 a 75 1736 a SDict begin 13 H.A end 75 1736 a 75 1736 a SDict begin [/View [/XYZ H.V]/Dest (Item.12) cvn /DEST pdfmark end 75 1736 a 85 x FF(Lemma)j(5.4.)240 b Fv(\(i\))41 b Fl(F)-7 b(or)39 b(every)e(automorphism)k Ft(')34 b Fr(2)g Fv(Aut\()p Ft(X)7 b Fv(\))38 b Fl(we)g(have)g Ft(b)23 b Fr(\016)h Ft(\013)p Fv(\()p Ft(')3175 1835 y Fo(\003)3216 1821 y Fv(\))34 b(=)g Ft(\013)p Fv(\()p Ft(')3542 1835 y Fo(\003)3583 1821 y Fv(\))24 b Fr(\016)g Ft(b)411 1937 y Fl(and)33 b(for)g Ft(n)25 b Fv(=)g(2)33 b Fl(also)1172 1859 y Fr(p)p 1248 1859 40 4 v 78 x Ft(b)20 b Fr(\016)h Ft(\013)p Fv(\()p Ft(')1525 1951 y Fo(\003)1565 1937 y Fv(\))26 b(=)f Ft(\013)p Fv(\()p Ft(')1874 1951 y Fo(\003)1915 1937 y Fv(\))20 b Fr(\016)2036 1859 y(p)p 2112 1859 V 78 x Ft(b)p Fl(.)75 1959 y SDict begin H.S end 75 1959 a 75 1959 a SDict begin 13 H.A end 75 1959 a 75 1959 a SDict begin [/View [/XYZ H.V]/Dest (Item.13) cvn /DEST pdfmark end 75 1959 a 248 2045 a Fv(\(ii\))41 b Fl(F)-7 b(or)31 b(every)g(line)f(bund)5 b(le)31 b Ft(L)25 b Fr(2)g Fv(Pic)o(\()p Ft(X)7 b Fv(\))32 b Fl(we)f(have)g Ft(b)16 b Fr(\016)g Ft(\013)p Fv(\()p Ft(M)2401 2059 y Fq(L)2454 2045 y Fv(\))26 b(=)f Ft(\013)p Fv(\()p Ft(M)2792 2059 y Fq(L)2845 2045 y Fv(\))16 b Fr(\016)h Ft(b)30 b Fl(and)i(for)f Ft(n)25 b Fv(=)g(2)31 b Fl(also)411 2083 y Fr(p)p 486 2083 40 4 v 486 2160 a Ft(b)21 b Fr(\016)f Ft(\013)p Fv(\()p Ft(M)792 2174 y Fq(L)845 2160 y Fv(\))26 b(=)f Ft(\013)p Fv(\()p Ft(M)1183 2174 y Fq(L)1236 2160 y Fv(\))c Fr(\016)1357 2083 y(p)p 1433 2083 V 77 x Ft(b)p Fl(.)75 2334 y(Pr)-5 b(o)g(of.)43 b Fv(W)-8 b(e)31 b(ha)m(v)m(e)g Ft(\013)p Fv(\()p Ft(')873 2348 y Fo(\003)914 2334 y Fv(\))20 b Fr(\016)g Ft(F)39 b Fr(')25 b Ft(F)32 b Fr(\016)20 b Ft(')1441 2348 y Fo(\003)1511 2334 y Fv(and)30 b Ft(\013)p Fv(\()p Ft(M)1869 2348 y Fq(L)1922 2334 y Fv(\))20 b Fr(\016)g Ft(F)38 b Fr(')25 b Ft(F)33 b Fr(\016)20 b Ft(M)2488 2301 y Fq(n)2478 2361 y(L)2535 2334 y Fv(.)40 b(The)30 b(assertions)g(no)m(w)g(follo)m(w)f(b)m(y)75 2454 y(lemma)373 2454 y SDict begin H.S end 373 2454 a Fv(2.3)489 2396 y SDict begin H.R end 489 2396 a 489 2454 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.3) cvn H.B /ANN pdfmark end 489 2454 a 36 w Fv(\(for)705 2377 y Fr(p)p 781 2377 40 4 v 77 x Ft(b)35 b Fv(one)h(has)g(to)g(use)g(the)f (analogous)h(result)f([)2344 2454 y SDict begin H.S end 2344 2454 a Fv(Ann07)2604 2392 y SDict begin H.R end 2604 2392 a 2604 2454 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Ann) cvn H.B /ANN pdfmark end 2604 2454 a Fv(,)j(prop)s(osition)33 b(2])j(for)g(spherical)75 2562 y(t)m(wists\).)3313 b Fj(\003)175 2753 y Fv(Let)31 b Ft(G)25 b Fr(\032)g Fv(Aut\(D)789 2717 y Fq(b)823 2753 y Fv(\()p Ft(X)940 2720 y Fp([)p Fq(n)p Fp(])1027 2753 y Fv(\)\))31 b(b)s(e)f(the)h(subgroup)d (generated)k(b)m(y)e Ft(b)p Fv(,)h(shifts,)e(and)h Ft(\013)p Fv(\(D)m(Aut)3196 2767 y Fq(st)3259 2753 y Fv(\()p Ft(X)7 b Fv(\)\).)75 2927 y SDict begin H.S end 75 2927 a 75 2927 a SDict begin 13 H.A end 75 2927 a 75 2927 a SDict begin [/View [/XYZ H.V]/Dest (theorem.5.5) cvn /DEST pdfmark end 75 2927 a FF(Prop)s(osition)36 b(5.5.)42 b Fl(The)33 b(map)310 3095 y Ft(S)15 b Fv(:)30 b Fs(Z)21 b Fr(\002)e Fs(Z)i Fr(\002)e Fv(\(Aut)q(\()p Ft(X)7 b Fv(\))21 b Ff(n)f Fv(Pic\()p Ft(X)7 b Fv(\)\))26 b Fr(!)g Fv(Aut\(D)1958 3058 y Fq(b)1958 3118 y Fn(S)2013 3126 y Fz(n)2060 3095 y Fv(\()p Ft(X)2177 3058 y Fq(n)2225 3095 y Fv(\)\))93 b Ft(;)109 b Fv(\()p Ft(k)s(;)15 b(`;)g Fv(\011\))26 b Fr(7!)g Ft(b)3013 3058 y Fq(k)3075 3095 y Fr(\016)21 b Fv([)p Ft(`)p Fv(])g Fr(\016)f Ft(\013)p Fv(\(\011\))75 3262 y Fl(de\014nes)33 b(a)g(gr)-5 b(oup)34 b(isomorphism)h(onto)f Ft(G)p Fl(.)75 3436 y(Pr)-5 b(o)g(of.)43 b Fv(By)32 b(the)f(previous)f (lemma,)i Ft(b)f Fv(indeed)f(comm)m(utes)j(with)d Ft(\013)p Fv(\(\011\))i(for)g(\011)27 b Fr(2)f Fv(D)m(Aut)3114 3450 y Fq(st)3177 3436 y Fv(\()p Ft(X)7 b Fv(\).)45 b(T)-8 b(ogether)75 3544 y(with)36 b(theorem)649 3544 y SDict begin H.S end 649 3544 a Fv(5.2)765 3485 y SDict begin H.R end 765 3485 a 765 3544 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.5.2) cvn H.B /ANN pdfmark end 765 3544 a 37 w Fv(and)g(the)h(fact)h(that)f(shifts)f(comm)m(ute)i(with)d(ev)m (ery)j(deriv)m(ed)e(automorphism,)h(this)75 3652 y(sho)m(ws)i(that)h Ft(S)k Fv(is)39 b(indeed)f(a)i(w)m(ell-de\014ned)d(group)i (homomorphism)e(with)h(image)i Ft(G)p Fv(.)68 b(No)m(w)40 b(consider)75 3775 y Ft(g)e Fv(=)33 b Ft(b)299 3742 y Fq(k)366 3775 y Fr(\016)24 b Fv([)p Ft(`)p Fv(])g Fr(\016)g Ft(\013)p Fv(\()p Ft(')768 3789 y Fo(\003)809 3775 y Fv(\))g Fr(\016)g Ft(\013)p Fv(\()p Ft(M)1118 3789 y Fq(L)1171 3775 y Fv(\))36 b(and)f(assume)h Ft(g)h Fv(=)d(id)o(.)56 b(F)-8 b(or)37 b(ev)m(ery)f(p)s(oin)m(t)f([)p Ft(\030)t Fv(])f Fr(2)g Ft(X)3048 3742 y Fp([)p Fq(n)p Fp(])3158 3775 y Fr(n)24 b Ft(X)3309 3727 y Fp([)p Fq(n)p Fp(])3302 3804 y(\001)3432 3775 y Fv(w)m(e)36 b(ha)m(v)m(e)75 3891 y Ft(g)s Fv(\()p Fs(C)p Fv(\([)p Ft(\030)t Fv(]\)\))46 b(=)d Fs(C)p Fv(\([)p Ft(')p Fv(\()p Ft(\030)t Fv(\)]\)[)p Ft(`)p Fv(])h(whic)m(h)c(sho)m(ws)h Ft(`)i Fv(=)g(0)e(and)g Ft(')j Fv(=)f(id)n(,)i(i.e.)c Ft(g)47 b Fv(=)c Ft(b)2864 3858 y Fq(k)2934 3891 y Fr(\016)28 b Ft(M)3095 3916 y Fq(L)3143 3897 y Fe(\002)p Fz(n)3234 3891 y Fv(.)73 b(Hence,)45 b(for)75 4006 y Ft(A)31 b Fr(2)f Fv(D)335 3970 y Fq(b)369 4006 y Fv(\()p Ft(X)7 b Fv(\))35 b(its)e(image)h(under)e Ft(F)47 b Fv(gets)34 b(mapp)s(ed)e(to)j Ft(g)s Fv(\()p Ft(F)13 b(A)p Fv(\))32 b(=)f Ft(F)13 b Fv(\()p Ft(A)23 b Fr(\012)f Ft(N)10 b Fv(\)[)p Ft(k)s Fv(\(2)p Ft(n)23 b Fr(\000)f Fv(2\)])35 b(for)f(some)g(line)75 4114 y(bundle)28 b Ft(N)40 b Fv(on)31 b Ft(X)7 b Fv(,)31 b(whic)m(h)e(sho)m(ws)h(that)h Ft(k)d Fv(=)d(0.)41 b(Finally)-8 b(,)29 b Ft(g)g Fv(=)c Ft(M)2334 4139 y Fq(L)2382 4120 y Fe(\002)p Fz(n)2504 4114 y Fv(is)k(trivial)g(only)g(if)g Ft(L)c Fv(=)g Fr(O)3403 4128 y Fq(X)3471 4114 y Fv(.)183 b Fj(\003)75 4298 y SDict begin H.S end 75 4298 a 75 4298 a SDict begin 13 H.A end 75 4298 a 75 4298 a SDict begin [/View [/XYZ H.V]/Dest (theorem.5.6) cvn /DEST pdfmark end 75 4298 a FF(Remark)34 b(5.6.)42 b Fv(Again,)31 b(the)f(analogous)h(statemen)m(t)h(with)d Ft(b)i Fv(replaced)f(b)m(y)2727 4220 y Fr(p)p 2803 4220 40 4 v 78 x Ft(b)g Fv(holds.)175 4472 y(Let)25 b(no)m(w)g Ft(X)33 b Fv(b)s(e)24 b(a)i(K3-surface.)39 b(In)24 b(this)g(case)j (Addington)c(has)i(sho)m(wn)g(in)e([)2766 4472 y SDict begin H.S end 2766 4472 a Fv(Add11)3027 4408 y SDict begin H.R end 3027 4408 a 3027 4472 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Add) cvn H.B /ANN pdfmark end 3027 4472 a Fv(])i(that)h(the)f(F)-8 b(ourier{)75 4585 y(Muk)j(ai)26 b(transform)g Ft(F)821 4599 y Fq(a)874 4585 y Fv(:)45 b(D)1014 4550 y Fq(b)1048 4585 y Fv(\()p Ft(X)7 b Fv(\))27 b Fr(!)e Fv(D)1412 4550 y Fq(b)1446 4585 y Fv(\()p Ft(X)1563 4552 y Fp([)p Fq(n)p Fp(])1650 4585 y Fv(\))j(with)d(k)m(ernel)i(the)g (univ)m(ersal)e(sheaf)i Fr(I)2992 4599 y Fp(\004)3069 4585 y Fv(is)f(a)h Fs(P)3300 4552 y Fq(n)p Fo(\000)p Fp(1)3464 4585 y Fv(functor)75 4699 y(with)37 b Ft(H)46 b Fv(=)38 b([)p Fr(\000)p Fv(2].)66 b(Here,)41 b(\004)d Fr(\032)g Ft(X)33 b Fr(\002)26 b Ft(X)1523 4666 y Fp([)p Fq(n)p Fp(])1648 4699 y Fv(is)37 b(the)i(univ)m(ersal)d(family)h(of)i (length)f Ft(n)f Fv(subsc)m(hemes.)65 b(W)-8 b(e)75 4807 y(denote)31 b(the)f(asso)s(ciated)h Fs(P)1028 4774 y Fq(n)p Fo(\000)p Fp(1)1166 4807 y Fv(-t)m(wist)f(b)m(y)g Ft(a)h Fv(and)f(in)f(case)i Ft(n)25 b Fv(=)g(2)31 b(the)f(spherical)f (t)m(wist)h(b)m(y)3240 4742 y Fr(p)p 3316 4742 49 4 v 65 x Ft(a)p Fv(.)75 4986 y SDict begin H.S end 75 4986 a 75 4986 a SDict begin 13 H.A end 75 4986 a 75 4986 a SDict begin [/View [/XYZ H.V]/Dest (theorem.5.7) cvn /DEST pdfmark end 75 4986 a FF(Lemma)d(5.7.)39 b Fl(F)-7 b(or)29 b(every)f(p)-5 b(oint)30 b Fv([)p Ft(\030)t Fv(])25 b Fr(2)g Ft(X)1555 4953 y Fp([)p Fq(n)p Fp(])1653 4986 y Fr(n)11 b Ft(@)5 b(X)1844 4953 y Fp([)p Fq(n)p Fp(])1931 4986 y Fl(,)29 b(i.e.)e Ft(\030)j Fv(=)25 b Fr(f)p Ft(x)2404 5000 y Fp(1)2443 4986 y Ft(;)15 b(:)g(:)g(:)i(;)e(x)2697 5000 y Fq(n)2744 4986 y Fr(g)2789 5001 y Fd(r)l(e)l(d)2914 4986 y Fl(with)29 b(p)-5 b(airwise)30 b(distinct)75 5100 y Ft(x)127 5114 y Fq(i)155 5100 y Fl(,)j(the)h(obje)-5 b(ct)33 b Ft(a)p Fv(\()p Fs(C)p Fv(\([)p Ft(\030)t Fv(]\)\))i Fl(is)f(supp)-5 b(orte)g(d)36 b(on)d(the)h(whole)g Ft(X)2138 5067 y Fp([)p Fq(n)p Fp(])2225 5100 y Fl(.)44 b(In)33 b(c)-5 b(ase)34 b Ft(n)26 b Fv(=)g(2)33 b Fl(the)h(same)g(holds)h(for)f (the)75 5208 y(obje)-5 b(ct)333 5142 y Fr(p)p 409 5142 49 4 v 66 x Ft(a)p Fv(\()p Fs(C)p Fv(\([)p Ft(\030)t Fv(]\)\))p Fl(.)p eop end end %%Page: 11 11 TeXDict begin HPSdict begin 11 10 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 67 -8 a SDict begin H.S end 67 -8 a 67 -8 a SDict begin H.R end 67 -8 a 67 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.11) cvn /DEST pdfmark end 67 -8 a 329 66 a Fp(NEW)30 b(DERIVED)h(A)n(UTOEQUIV)-8 b(ALENCES)30 b(OF)h(HILBER)-6 b(T)31 b(SCHEMES)f(AND)g(KUMMER)g(V)-8 b(ARIETIES)183 b(11)75 266 y Fl(Pr)-5 b(o)g(of.)43 b Fv(W)-8 b(e)22 b(set)f(for)f(short)g Ft(A)26 b Fv(=)e Fs(C)p Fv(\([)p Ft(\030)t Fv(]\).)39 b(W)-8 b(e)22 b(will)c(use)i(the)g(exact)i (triangle)e(of)h(F)-8 b(ourier{Muk)j(ai)20 b(transforms)75 374 y Ft(F)42 b Fr(!)29 b Ft(F)366 341 y Fo(0)418 374 y Fr(!)g Ft(F)609 341 y Fo(00)684 374 y Fv(with)j(k)m(ernels)g Fr(P)1270 341 y Fo(0)1323 374 y Fv(=)c Fr(O)1494 399 y Fq(X)5 b Fo(\002)p Fq(X)1675 380 y Fx([)p Fz(n)p Fx(])1790 374 y Fv(and)32 b Fr(P)2039 341 y Fo(00)2111 374 y Fv(=)d Fr(O)2283 388 y Fp(\004)2335 374 y Fv(.)47 b(The)32 b(righ)m(t)g (adjoin)m(ts)g(form)h(the)f(exact)75 484 y(triangle)e Ft(R)479 451 y Fo(00)547 484 y Fr(!)c Ft(R)734 451 y Fo(0)783 484 y Fr(!)g Ft(R)32 b Fv(with)d(k)m(ernels)i Fr(Q)1587 451 y Fo(00)1656 484 y Fv(=)25 b Fr(O)1827 451 y Fo(_)1824 511 y Fp(\004)1878 484 y Fv([2])32 b(and)e Fr(Q)2256 451 y Fo(0)2306 484 y Fv(=)c Fr(O)2475 510 y Fq(X)5 b Fo(\002)p Fq(X)2656 490 y Fx([)p Fz(n)p Fx(])2738 484 y Fv([2].)43 b(Ov)m(er)31 b(the)g(op)s(en)g(subset)75 608 y Ft(X)157 575 y Fp([)p Fq(n)p Fp(])259 608 y Fr(n)15 b Ft(@)5 b(X)454 575 y Fp([)p Fq(n)p Fp(])541 608 y Fv(,)28 b(the)g(univ)m(ersal)e(family)g(\004)i(is)f(smo)s(oth)g(and)g(th)m(us)h (on)f(\004)2457 634 y Fo(j)p Fq(X)2540 614 y Fx([)p Fz(n)p Fx(])2617 634 y Fo(n)p Fq(@)t(X)2756 614 y Fx([)p Fz(n)p Fx(])2866 608 y Fv(the)h(ob)5 b(ject)29 b Fr(O)3368 575 y Fo(_)3365 635 y Fp(\004)3447 608 y Fv(is)d(a)j(line)75 722 y(bundle)f(concen)m(trated)k(in)d(degree)i(2.)41 b(This)29 b(yields)736 885 y Ft(R)806 848 y Fo(00)848 885 y Fv(\()p Ft(A)p Fv(\))d(=)f Fr(O)1180 900 y Fq(\030)1218 885 y Fv([0])92 b Ft(;)106 b(R)1606 848 y Fo(0)1629 885 y Fv(\()p Ft(A)p Fv(\))27 b(=)e(H)1958 848 y Fo(\003)1997 885 y Fv(\()p Ft(X)2114 848 y Fp([)p Fq(n)p Fp(])2201 885 y Ft(;)15 b(A)p Fv(\))21 b Fr(\012)f(O)2528 899 y Fq(X)2596 885 y Fv([2])26 b(=)f Fr(O)2885 899 y Fq(X)2953 885 y Fv([2])15 b Ft(:)75 1051 y Fv(Setting)22 b Ft(H)463 1018 y Fq(i)516 1051 y Fv(=)j Fr(H)690 1015 y Fq(i)718 1051 y Fv(\()p Ft(R)q Fv(\()p Ft(A)p Fv(\)\))e(the)f(long)g(exact)h (cohomology)g(sequence)g(giv)m(es)f Ft(H)2742 1018 y Fo(\000)p Fp(2)2862 1051 y Fv(=)j Fr(O)3030 1065 y Fq(X)3097 1051 y Fv(,)f Ft(H)3229 1018 y Fo(\000)p Fp(1)3349 1051 y Fv(=)h Fr(O)3517 1066 y Fq(\030)3555 1051 y Fv(,)f(and)75 1164 y Ft(H)158 1131 y Fq(i)211 1164 y Fv(=)h(0)c(for)g(all)e(other)i (v)-5 b(alues)20 b(of)g Ft(i)p Fv(.)38 b(The)20 b(only)g(functor)g(in)f (the)i(comp)s(osition)e Ft(F)38 b Fv(=)25 b(pr)2987 1189 y Fq(X)3050 1170 y Fx([)p Fz(n)p Fx(])3128 1189 y Fo(\003)3183 1164 y Fv(\(pr)3304 1131 y Fo(\003)3304 1191 y Fq(X)3372 1164 y Fv(\()p 3407 1179 61 4 v 60 w(\))c Fr(\012)f(I)3664 1178 y Fp(\004)3715 1164 y Fv(\))75 1272 y(that)28 b(needs)f(to)h(b)s (e)f(deriv)m(ed)g(is)f(the)i(push-forw)m(ard)d(along)j(pr)2175 1297 y Fq(X)2238 1278 y Fx([)p Fz(n)p Fx(])2320 1272 y Fv(.)39 b(The)27 b(reason)h(is)e(that)i(the)g(non-deriv)m(ed)75 1380 y(functors)j(pr)515 1347 y Fo(\003)515 1407 y Fq(X)614 1380 y Fv(as)h(w)m(ell)f(as)h(pr)1113 1347 y Fo(\003)1113 1407 y Fq(X)1180 1380 y Fv(\()p 1215 1395 V 61 w(\))22 b Fr(\012)f(O)1497 1394 y Fp(\004)1580 1380 y Fv(are)33 b(exact)g(\(see)g([)2184 1380 y SDict begin H.S end 2184 1380 a Fv(Sca09b)2462 1317 y SDict begin H.R end 2462 1317 a 2462 1380 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Sca2) cvn H.B /ANN pdfmark end 2462 1380 a Fv(,)g(prop)s(osition)d (2.3])j(for)f(the)g(latter\).)75 1488 y(Th)m(us,)e(there)g(is)g(the)g (sp)s(ectral)g(sequence)1087 1646 y Ft(E)1159 1601 y Fq(p;q)1154 1672 y Fp(2)1278 1646 y Fv(=)25 b Fr(H)1452 1608 y Fq(p)1492 1646 y Fv(\()p Ft(F)13 b Fv(\()p Ft(H)1716 1608 y Fq(q)1754 1646 y Fv(\)\))26 b(=)-15 b Fr(\))25 b Ft(E)2094 1608 y Fq(n)2167 1646 y Fv(=)g Fr(H)2340 1608 y Fq(n)2387 1646 y Fv(\()p Ft(F)13 b(R)q Fv(\()p Ft(A)p Fv(\)\))75 1804 y(asso)s(ciated)41 b(to)g(the)f(deriv)m(ed)f (functor)h(pr)1545 1829 y Fq(X)1608 1810 y Fx([)p Fz(n)p Fx(])1686 1829 y Fo(\003)1725 1804 y Fv(.)70 b(It)41 b(is)e(zero)i(outside)e(of)i(the)f Fr(\000)p Fv(1)g(and)g Fr(\000)p Fv(2)g(ro)m(w.)70 b(No)m(w)75 1916 y Ft(F)146 1883 y Fo(0)169 1916 y Fv(\()p Fr(O)276 1931 y Fq(\030)315 1916 y Fv(\))25 b(=)g(H)540 1880 y Fo(\003)579 1916 y Fv(\()p Ft(X)r(;)15 b Fr(O)803 1931 y Fq(\030)842 1916 y Fv(\))20 b Fr(\012)e(O)1058 1941 y Fq(X)1121 1922 y Fx([)p Fz(n)p Fx(])1229 1916 y Fv(=)25 b Fr(O)1400 1878 y Fo(\010)p Fq(n)1397 1956 y(X)1460 1936 y Fx([)p Fz(n)p Fx(])1542 1916 y Fv([0])31 b(and)e Ft(F)1915 1883 y Fo(00)1958 1916 y Fv(\()p Fr(O)2065 1931 y Fq(\030)2103 1916 y Fv(\))h(is)f(also)h (concen)m(trated)h(in)e(degree)h(zero)h(since)75 2047 y(\004)g(is)f(\014nite)h(o)m(v)m(er)i Ft(X)771 2014 y Fp([)p Fq(n)p Fp(])857 2047 y Fv(.)44 b(By)32 b(the)g(long)f(exact)i (sequence)e(w)m(e)h(see)g(that)g(all)f(terms)g(in)f(the)i Fr(\000)p Fv(1)f(ro)m(w)h(except)75 2160 y(for)e Ft(E)286 2116 y Fp(0)p Fq(;)p Fo(\000)p Fp(1)281 2186 y(2)466 2160 y Fv(and)g Ft(E)715 2116 y Fp(1)p Fq(;)p Fo(\000)p Fp(1)710 2186 y(2)895 2160 y Fv(m)m(ust)g(v)-5 b(anish.)39 b(F)-8 b(urthermore,)834 2325 y Ft(F)905 2287 y Fo(0)928 2325 y Fv(\()p Ft(H)1046 2287 y Fo(\000)p Fp(2)1141 2325 y Fv(\))25 b(=)g(H)1366 2287 y Fo(\003)1405 2325 y Fv(\()p Ft(X)r(;)15 b Fr(O)1629 2339 y Fq(X)1698 2325 y Fv(\))20 b Fr(\012)g(O)1916 2351 y Fq(X)1979 2331 y Fx([)p Fz(n)p Fx(])2087 2325 y Fv(=)25 b Fr(O)2255 2351 y Fq(X)2318 2331 y Fx([)p Fz(n)p Fx(])2400 2325 y Fv([0])c Fr(\010)f(O)2679 2351 y Fq(X)2742 2331 y Fx([)p Fz(n)p Fx(])2825 2325 y Fv([)p Fr(\000)p Fv(2])75 2485 y(and)35 b Ft(F)328 2452 y Fo(00)370 2485 y Fv(\()p Ft(H)488 2452 y Fo(\000)p Fp(2)583 2485 y Fv(\))g(is)g(a)h(lo)s(cally)d(free)j(sheaf)f(of)h(rank) e Ft(n)h Fv(concen)m(trated)i(in)d(degree)i(zero)g(since)f(\004)g(is)f (\015at)i(of)75 2598 y(degree)f Ft(n)f Fv(o)m(v)m(er)h Ft(X)731 2565 y Fp([)p Fq(n)p Fp(])818 2598 y Fv(.)52 b(This)32 b(sho)m(ws)i(that)h(the)g Fr(\000)p Fv(2)f(ro)m(w)g(of)h Ft(E)2236 2612 y Fp(2)2309 2598 y Fv(is)f(zero)h(outside)e(of)h(degree) h(0,)h(1,)g(and)e(2)75 2711 y(and)g(that)g Ft(E)528 2667 y Fp(1)p Fq(;)p Fo(\000)p Fp(2)523 2738 y(2)712 2711 y Fv(is)f(of)h(p)s(ositiv)m(e)f(rank.)52 b(By)34 b(the)h(p)s (ositioning)c(of)j(the)h(non-zero)f(terms)g(it)g(follo)m(ws)f(that)75 2833 y Ft(E)147 2789 y Fp(1)p Fq(;)p Fo(\000)p Fp(2)142 2859 y(2)329 2833 y Fv(=)e Ft(E)503 2789 y Fp(1)p Fq(;)p Fo(\000)p Fp(2)498 2845 y Fo(1)687 2833 y Fv(and)j(th)m(us)g(also)h Ft(E)1331 2800 y Fo(\000)p Fp(1)1457 2833 y Fv(=)d Fr(H)1638 2797 y Fo(\000)p Fp(1)1732 2833 y Fv(\()p Ft(F)13 b(R)q Fv(\()p Ft(A)p Fv(\)\))36 b(is)e(of)g(p)s(ositiv)m(e)g(rank.)52 b(F)-8 b(urthermore,)36 b(w)m(e)f(can)75 2941 y(read)d(o\013)g(the)h (sp)s(ectral)e(sequence)h(that)h(the)f(cohomology)i(of)e Ft(F)13 b(R)q Fv(\()p Ft(A)p Fv(\))32 b(is)g(concen)m(trated)h(in)e (the)h(degrees)75 3049 y Fr(\000)p Fv(2,)37 b Fr(\000)p Fv(1,)h(and)d(0.)57 b(No)m(w,)38 b(b)m(y)e(the)g(long)f(exact)j (sequences)e(asso)s(ciated)g(to)h(the)f(cones)g(o)s(ccurring)e(in)h (the)75 3157 y(de\014nition)h(of)j(the)g(spherical)d(resp)s(ectiv)m (ely)i(the)h Fs(P)1891 3124 y Fq(n)1938 3157 y Fv(-t)m(wist)g(it)f (follo)m(ws)f(that)j Fr(H)2890 3121 y Fo(\000)p Fp(2)2985 3157 y Fv(\()3020 3091 y Fr(p)p 3096 3091 49 4 v 66 x Ft(a)p Fv(\()p Ft(A)p Fv(\)\))f(as)g(w)m(ell)f(as)75 3266 y Fr(H)153 3230 y Fo(\000)p Fp(2)247 3266 y Fv(\()p Ft(a)p Fv(\()p Ft(A)p Fv(\)\))32 b(are)f(of)f(p)s(ositiv)m(e)g(rank.) 2347 b Fj(\003)75 3350 y SDict begin H.S end 75 3350 a 75 3350 a SDict begin 13 H.A end 75 3350 a 75 3350 a SDict begin [/View [/XYZ H.V]/Dest (Item.14) cvn /DEST pdfmark end 75 3350 a 85 x FF(Prop)s(osition)36 b(5.8.)240 b Fv(\(i\))42 b Fl(The)34 b(sub)-5 b(gr)g(oup)36 b Ft(H)41 b Fl(gener)-5 b(ate)g(d)36 b(by)e Ft(a)g Fl(and)h(push-forwar)-5 b(ds)38 b(along)d(natur)-5 b(al)411 3564 y(automorphisms,)52 b(i.e.)45 b(auto)-5 b(e)g(quivalenc)g(es)47 b(of)f(the)g(form)h Ft(')2524 3516 y Fp([)p Fq(n)p Fp(])2524 3578 y Fo(\003)2659 3564 y Fv(=)i Ft(\013)p Fv(\()p Ft(')2931 3578 y Fo(\003)2972 3564 y Fv(\))p Fl(,)g(is)c(isomorphic)j(to)411 3672 y Fs(Z)20 b Fr(\002)g Fv(Aut\()p Ft(X)7 b Fv(\))p Fl(.)75 3695 y SDict begin H.S end 75 3695 a 75 3695 a SDict begin 13 H.A end 75 3695 a 75 3695 a SDict begin [/View [/XYZ H.V]/Dest (Item.15) cvn /DEST pdfmark end 75 3695 a 248 3801 a Fv(\(ii\))41 b Ft(b)35 b(=)-55 b Fr(2)25 b Ft(H)32 b Fv(=)25 b Fr(h)p Ft(a;)15 b Fr(f)p Ft(')992 3753 y Fp([)p Fq(n)p Fp(])992 3815 y Fo(\003)1080 3801 y Fr(g)1125 3819 y Fq(')p Fo(2)p Fp(Aut)q(\()p Fq(X)5 b Fp(\))1460 3801 y Fr(i)p Fl(.)75 3836 y SDict begin H.S end 75 3836 a 75 3836 a SDict begin 13 H.A end 75 3836 a 75 3836 a SDict begin [/View [/XYZ H.V]/Dest (Item.16) cvn /DEST pdfmark end 75 3836 a 223 3912 a Fv(\(iii\))40 b Ft(a)35 b(=)-55 b Fr(2)25 b Ft(G)g Fv(=)g Fr(h)p Ft(b;)15 b Fv([)p Ft(`)p Fv(])p Ft(;)g(\013)p Fv(\(D)m(Aut)1321 3926 y Fq(st)1383 3912 y Fv(\()p Ft(X)7 b Fv(\)\))p Fr(i)p Fl(.)175 4028 y(The)32 b(same)i(r)-5 b(esults)33 b(hold)i(for)e Ft(a)f Fl(r)-5 b(eplac)g(e)g(d)35 b(by)1762 3963 y Fr(p)p 1838 3963 49 4 v 65 x Ft(a)d Fl(and)i Ft(b)e Fl(r)-5 b(eplac)g(e)g(d)35 b(by)2630 3951 y Fr(p)p 2705 3951 40 4 v 2705 4028 a Ft(b)q Fl(.)75 4217 y(Pr)-5 b(o)g(of.)43 b Fv(W)-8 b(e)35 b(ha)m(v)m(e)h(for)e Ft(')e Fr(2)g Fv(Aut\()p Ft(X)7 b Fv(\))35 b(that)g Ft(')1658 4169 y Fp([)p Fq(n)p Fp(])1658 4231 y Fo(\003)1767 4217 y Fr(\016)24 b Ft(F)1894 4231 y Fq(a)1968 4217 y Fv(=)31 b Ft(F)2128 4231 y Fq(a)2193 4217 y Fr(\016)23 b Ft(')2320 4231 y Fo(\003)2395 4217 y Fv(whic)m(h)33 b(b)m(y)h(lemma)3086 4217 y SDict begin H.S end 3086 4217 a Fv(2.3)3202 4158 y SDict begin H.R end 3202 4158 a 3202 4217 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.3) cvn H.B /ANN pdfmark end 3202 4217 a 34 w Fv(sho)m(ws)g(that)h Ft(a)75 4346 y Fv(comm)m(utes)f(with)d Ft(')773 4298 y Fp([)p Fq(n)p Fp(])773 4360 y Fo(\003)860 4346 y Fv(.)48 b(The)32 b(reason)h(is)f(that)h(the)g(sub)m(v)-5 b(ariet)m(y)33 b(\004)c Fr(\032)g Ft(X)g Fr(\002)21 b Ft(X)2766 4313 y Fp([)p Fq(n)p Fp(])2886 4346 y Fv(is)32 b(in)m(v)-5 b(arian)m(t)31 b(under)h(the)75 4469 y(morphism)22 b Ft(')8 b Fr(\002)g Ft(')699 4436 y Fp([)p Fq(n)p Fp(])785 4469 y Fv(.)38 b(Because)26 b(of)e Ft(a)1335 4436 y Fq(k)1385 4469 y Fr(\016)8 b Ft(')1497 4421 y Fp([)p Fq(n)p Fp(])1497 4483 y Fo(\003)1584 4469 y Fv(\()p Ft(F)1677 4483 y Fq(a)1719 4469 y Fv(\()p Ft(A)p Fv(\)\))27 b(=)e Ft(F)2073 4483 y Fq(a)2115 4469 y Fv(\()p Ft(')2209 4436 y Fq(k)2209 4492 y Fo(\003)2252 4469 y Ft(A)p Fv(\)[)p Ft(k)s Fv(2\()p Ft(n)8 b Fr(\000)g Fv(1\)])26 b(for)e Ft(A)h Fr(2)g Fv(D)3164 4433 y Fq(b)3199 4469 y Fv(\()p Ft(X)7 b Fv(\),)26 b(there)f(are)75 4598 y(no)31 b(further)g(relations)f(in)h(the)g(group)g Ft(H)39 b Fv(whic)m(h)30 b(sho)m(ws)i(\(i\).)44 b(The)31 b(auto)s(equiv)-5 b(alence)31 b Ft(g)g Fv(=)c Ft(a)3276 4565 y Fq(k)3340 4598 y Fr(\016)21 b Ft(')3465 4550 y Fp([)p Fq(n)p Fp(])3465 4612 y Fo(\003)3579 4598 y Fr(2)27 b Ft(H)75 4706 y Fv(has)22 b Ft(g)s Fv(\()p Ft(F)13 b Fv(\()p Fr(O)488 4720 y Fq(X)558 4706 y Fv(\)\))26 b(=)f Ft(F)13 b Fv(\()p Fr(O)928 4720 y Fq(X)996 4706 y Fv(\)[2)p Ft(k)s Fv(\()p Ft(n)5 b Fr(\000)g Fv(1\)].)40 b(Th)m(us,)23 b(b)m(y)g(remark)2163 4706 y SDict begin H.S end 2163 4706 a Fv(2.6)2279 4647 y SDict begin H.R end 2279 4647 a 2279 4706 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.6) cvn H.B /ANN pdfmark end 2279 4706 a 23 w Fv(the)g(equalit)m(y)f Ft(b)k Fv(=)f Ft(g)h Fv(implies)20 b Ft(k)28 b Fv(=)d(1.)39 b(But)75 4835 y(also)23 b Ft(b)i Fv(=)g Ft(a)6 b Fr(\016)g Ft(')574 4787 y Fp([)p Fq(n)p Fp(])574 4849 y Fo(\003)684 4835 y Fv(can)23 b(not)h(hold)e(comparing)g(the)h(v)-5 b(alues)23 b(of)g(b)s(oth)g(sides)f(on)h Fs(C)p Fv(\([)p Ft(\030)t Fv(]\))h(for)f([)p Ft(\030)t Fv(])i Fr(2)g Ft(X)3359 4802 y Fp([)p Fq(n)p Fp(])3452 4835 y Fr(n)6 b Ft(@)f(X)3638 4802 y Fp([)p Fq(n)p Fp(])3725 4835 y Fv(.)75 4943 y(The)21 b(assertion)g(\(iii\))f(also)h(is)g(sho)m(wn)g(b) m(y)g(comparing)g(the)h(v)-5 b(alues)20 b(of)i(the)g(auto)s(equiv)-5 b(alences)21 b(on)g Fs(C)p Fv(\([)p Ft(\030)t Fv(]\).)92 b Fj(\003)175 5117 y Fv(Using)42 b(the)i(same)g(argumen)m(ts)g(as)g(in) e([)1579 5117 y SDict begin H.S end 1579 5117 a Fv(Add11)1839 5053 y SDict begin H.R end 1839 5053 a 1839 5117 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Add) cvn H.B /ANN pdfmark end 1839 5117 a Fv(,)48 b(p.11-12)d(and)e(p.39-40])j (one)d(can)h(also)g(sho)m(w)f(that)75 5230 y Ft(b)h Fv(do)s(es)f(not)h (equal)g(a)g(shift)f(of)h(an)f(auto)s(equiv)-5 b(alence)44 b(induced)e(b)m(y)i(a)g Fs(P)2694 5197 y Fq(n)2741 5230 y Fv(-ob)5 b(ject)45 b(on)f Ft(X)3282 5197 y Fp([)p Fq(n)p Fp(])3412 5230 y Fv(or)g(of)g(an)p eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 67 -8 a SDict begin H.S end 67 -8 a 67 -8 a SDict begin H.R end 67 -8 a 67 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.12) cvn /DEST pdfmark end 67 -8 a 75 66 a Fp(12)1473 b(ANDREAS)24 b(KR)n(UG)75 266 y Fv(auto)s(equiv)-5 b(alence)30 b(of)g(the)g(form)f Ft(\013)p Fv(\()p Ft(T)1353 280 y Fq(E)1414 266 y Fv(\))h(for)f(a)i(spherical)d(t)m(wist)i Ft(T)2350 280 y Fq(E)2439 266 y Fv(on)g(the)g(surface.)40 b(In)30 b(particular,)e Ft(b)i Fv(is)75 374 y(an)g(exotic)h(auto)s (equiv)-5 b(alence)31 b(in)e(the)h(sense)g(of)h([)1750 374 y SDict begin H.S end 1750 374 a Fv(PS12)1954 311 y SDict begin H.R end 1954 311 a 1954 374 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.PS) cvn H.B /ANN pdfmark end 1954 374 a Fv(].)75 586 y SDict begin H.S end 75 586 a 75 586 a SDict begin 13 H.A end 75 586 a 75 586 a SDict begin [/View [/XYZ H.V]/Dest (section.6) cvn /DEST pdfmark end 75 586 a 812 672 a Fv(6.)46 b Fs(P)999 639 y Fq(n)1046 672 y Fu(-objects)33 b(on)h(generalised)e(Kummer)h(v)-8 b(arieties)175 834 y Fv(Let)31 b Ft(A)f Fv(b)s(e)g(an)g(ab)s(elian)f (surface.)40 b(There)30 b(is)g(the)g(summation)f(map)1143 1089 y(\006)10 b(:)31 b Ft(A)1343 1051 y Fq(n)1415 1089 y Fr(!)25 b Ft(A)91 b(;)107 b Fv(\()p Ft(a)1905 1103 y Fp(1)1945 1089 y Ft(;)15 b(:)g(:)g(:)h(;)f(a)2194 1103 y Fq(n)2242 1089 y Fv(\))25 b Fr(7!)2463 975 y Fq(n)2418 1002 y Fk(X)2427 1198 y Fq(i)p Fp(=1)2565 1089 y Ft(a)2613 1103 y Fq(i)2656 1089 y Ft(:)75 1369 y Fv(W)-8 b(e)32 b(set)e Ft(N)446 1383 y Fq(n)p Fo(\000)p Fp(1)584 1369 y Ft(A)25 b Fv(:=)g(\006)864 1336 y Fo(\000)p Fp(1)958 1369 y Fv(\(0\).)42 b(It)31 b(is)e(isomorphic)f(to)j Ft(A)1969 1336 y Fq(n)p Fo(\000)p Fp(1)2137 1369 y Fv(via)f(e.g.)i(the) e(morphism)677 1643 y Ft(A)745 1606 y Fq(n)p Fo(\000)p Fp(1)908 1643 y Fr(!)25 b Ft(N)1097 1657 y Fq(n)p Fo(\000)p Fp(1)1234 1643 y Ft(A)91 b(;)107 b Fv(\()p Ft(a)1608 1657 y Fp(1)1647 1643 y Ft(;)15 b(:)g(:)g(:)i(;)e(a)1897 1657 y Fq(n)p Fo(\000)p Fp(1)2035 1643 y Fv(\))25 b Fr(7!)g Fv(\()p Ft(a)2294 1657 y Fp(1)2334 1643 y Ft(;)15 b(:)g(:)g(:)i(;)e(a) 2584 1657 y Fq(n)p Fo(\000)p Fp(1)2721 1643 y Ft(;)g Fr(\000)2847 1530 y Fq(n)p Fo(\000)p Fp(1)2849 1557 y Fk(X)2857 1752 y Fq(i)p Fp(=1)2996 1643 y Ft(a)3044 1657 y Fq(i)3072 1643 y Fv(\))g Ft(:)75 1916 y Fv(The)30 b(sub)m(v)-5 b(ariet)m(y)30 b Ft(N)770 1930 y Fq(n)p Fo(\000)p Fp(1)908 1916 y Ft(A)25 b Fr(\032)h Ft(A)1166 1883 y Fq(n)1243 1916 y Fv(is)k Fm(S)1410 1930 y Fq(n)1457 1916 y Fv(-in)m(v)-5 b(arian)m(t.)41 b(Th)m(us,)30 b(w)m(e)h(ha)m(v)m(e)h Ft(N)2578 1930 y Fq(n)p Fo(\000)p Fp(1)2715 1916 y Ft(A=)p Fm(S)2903 1930 y Fq(n)2977 1916 y Fr(\032)25 b Ft(S)3134 1883 y Fq(n)3181 1916 y Ft(A)p Fv(.)41 b(The)31 b Fl(gener-)75 2024 y(alise)-5 b(d)28 b(Kummer)f(variety)33 b Fv(is)23 b(de\014ned)f(as)i Ft(K)1570 2038 y Fq(n)p Fo(\000)p Fp(1)1708 2024 y Ft(A)h Fv(:=)g Ft(\026)1977 1991 y Fo(\000)p Fp(1)2071 2024 y Fv(\()p Ft(N)2179 2038 y Fq(n)p Fo(\000)p Fp(1)2317 2024 y Ft(A=)p Fm(S)2505 2038 y Fq(n)2553 2024 y Fv(\),)h(i.e.)e(it)f(is)g(the)h(sub)m(v)-5 b(ariet)m(y)24 b(of)g(the)75 2138 y(Hilb)s(ert)k(sc)m(heme)i Ft(A)766 2105 y Fp([)p Fq(n)p Fp(])882 2138 y Fv(consisting)e(of)i(all)f(p)s (oin)m(ts)f(represen)m(ting)h(subsc)m(hemes)g(whose)h(w)m(eigh)m(ted)g (supp)s(ort)75 2246 y(adds)g(up)g(to)h(zero.)43 b(It)31 b(can)g(b)s(e)f(iden)m(ti\014ed)f(with)g(Hilb)1923 2209 y Fn(S)1978 2217 y Fz(n)2025 2246 y Fv(\()p Ft(N)2133 2260 y Fq(n)p Fo(\000)p Fp(1)2271 2246 y Ft(A)p Fv(\))i(and)f(also)h (the)g(other)g(assumptions)e(of)75 2354 y(the)i(Bridgeland{King{Reid)c (theorem)k(are)g(satis\014ed)e(whic)m(h)g(leads)h(to)h(the)g(equiv)-5 b(alence)1056 2542 y(\010)25 b(=)g(FM)1386 2556 y Fo(O)p 1442 2538 124 3 v 25 x Fz(I)1473 2567 y(n)1515 2581 y(A)1580 2542 y Fv(:)45 b(D)1720 2504 y Fq(b)1754 2542 y Fv(\()p Ft(K)1866 2556 y Fq(n)p Fo(\000)p Fp(1)2004 2542 y Ft(A)p Fv(\))26 b Fr(!)f Fv(D)2318 2504 y Fq(b)2318 2565 y Fn(S)2373 2573 y Fz(n)2420 2542 y Fv(\()p Ft(N)2528 2556 y Fq(n)p Fo(\000)p Fp(1)2665 2542 y Ft(A)p Fv(\))75 2747 y(where)p 342 2674 177 4 v 34 w Ft(I)389 2721 y Fq(n)436 2747 y Ft(X)39 b Fv(=)32 b Ft(p)699 2714 y Fo(\000)p Fp(1)793 2747 y Fv(\()p Ft(N)901 2761 y Fq(n)p Fo(\000)p Fp(1)1038 2747 y Ft(A)p Fv(\))j(\(see)g(e.g.)h([)1558 2747 y SDict begin H.S end 1558 2747 a Fv(Nam02)1838 2685 y SDict begin H.R end 1838 2685 a 1838 2747 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Nam) cvn H.B /ANN pdfmark end 1838 2747 a Fv(]\).)54 b(The)33 b(in)m(tersection)h(b)s(et)m(w)m(een)h(the)g (small)e(diagonal)75 2855 y(\001)44 b(=)g Ft(\016)s Fv(\()p Ft(A)p Fv(\))i Fr(\032)e Ft(A)720 2822 y Fq(n)809 2855 y Fv(and)d Ft(N)1070 2869 y Fq(n)p Fo(\000)p Fp(1)1207 2855 y Ft(A)h Fv(consists)f(exactly)i(of)e(the)h(p)s(oin)m(ts)f Ft(\016)s Fv(\()p Ft(a)p Fv(\))46 b(=)e(\()p Ft(a;)15 b(:)g(:)g(:)i(;)e(a)p Fv(\))42 b(for)g Ft(a)f Fv(an)h Ft(n)p Fv(-)75 2963 y(torsion)e(p)s(oin)m(t)g(of)g Ft(A)p Fv(,)k(i.e.)c(\001)27 b Fr(\\)g Ft(N)1307 2977 y Fq(n)p Fo(\000)p Fp(1)1444 2963 y Ft(A)42 b Fv(=)g Ft(\016)s Fv(\()p Ft(A)1813 2977 y Fq(n)1862 2963 y Fv(\).)71 b(The)40 b(in)m(tersection)h(is)e(transv)m(ersal,)44 b(since)39 b(under)75 3072 y(the)e(iden)m(ti\014cation)e Ft(T)845 3086 y Fq(A)898 3068 y Fz(n)981 3047 y Fr(\030)981 3077 y Fv(=)1087 3072 y Ft(T)1153 3034 y Fo(\010)p Fq(n)1140 3101 y(A)1291 3072 y Fv(the)i(tangen)m(t)h(space)f(of)g(\001)g(in)e(a)i (p)s(oin)m(t)f Ft(\016)s Fv(\()p Ft(a)p Fv(\))i(with)d Ft(a)h Fr(2)f Ft(A)3361 3086 y Fq(n)3445 3072 y Fv(is)h(giv)m(en)75 3182 y(b)m(y)j(v)m(ectors)h(of)f(the)h(form)e(\()p Ft(v)s(;)15 b(:)g(:)g(:)i(;)e(v)s Fv(\))41 b Fr(2)e Ft(T)1588 3196 y Fq(A)1645 3182 y Fv(\()p Ft(a)p Fv(\))1763 3149 y Fo(\010)p Fq(n)1905 3182 y Fv(whereas)g(the)g(tangen)m(t)h(space)g(of)f Ft(N)3199 3196 y Fq(n)p Fo(\000)p Fp(1)3336 3182 y Ft(A)g Fv(is)f(giv)m(en)75 3290 y(b)m(y)f(v)m(ectors)i(\()p Ft(v)603 3304 y Fp(1)643 3290 y Ft(;)15 b(:)g(:)g(:)i(;)e(v)889 3304 y Fq(n)936 3290 y Fv(\))38 b Fr(2)e Ft(T)1159 3304 y Fq(A)1216 3290 y Fv(\()p Ft(a)p Fv(\))1334 3257 y Fo(\010)p Fq(n)1474 3290 y Fv(with)1689 3222 y Fk(P)1785 3249 y Fq(n)1785 3317 y(i)p Fp(=1)1918 3290 y Ft(v)1962 3304 y Fq(i)2027 3290 y Fv(=)h(0.)63 b(Th)m(us,)38 b(w)m(e)g(ha)m(v)m(e)h (for)e(the)g(tangen)m(t)i(space)75 3398 y(of)h Ft(N)261 3412 y Fq(n)p Fo(\000)p Fp(1)398 3398 y Ft(A)g Fv(in)e Ft(\016)s Fv(\()p Ft(a)p Fv(\))j(the)f(iden)m(ti\014cation)e Ft(T)1599 3412 y Fq(N)1655 3421 y Fz(n)p Fy(\000)p Fx(1)1776 3412 y Fq(A)1833 3398 y Fv(\()p Ft(\016)s Fv(\()p Ft(a)p Fv(\)\))2107 3373 y Fr(\030)2107 3402 y Fv(=)2218 3398 y Ft(N)2291 3417 y Fp(\001)p Fq(=)l(A)2434 3398 y Fz(n)9 b Fv(\()p Ft(\016)s Fv(\()p Ft(a)p Fv(\)\).)70 b(Since)39 b(the)h Fm(S)3295 3412 y Fq(n)3342 3398 y Fv(-action)g(on)75 3506 y Ft(N)148 3520 y Fq(n)p Fo(\000)p Fp(1)285 3506 y Ft(A)31 b Fv(is)e(just)h(the)g(restriction)g(of)g(the)h(action)g(on)f Ft(A)1974 3473 y Fq(n)2021 3506 y Fv(,)h(this)e(isomorphism)e(is)j (equiv)-5 b(arian)m(t.)75 3691 y SDict begin H.S end 75 3691 a 75 3691 a SDict begin 13 H.A end 75 3691 a 75 3691 a SDict begin [/View [/XYZ H.V]/Dest (theorem.6.1) cvn /DEST pdfmark end 75 3691 a FF(Theorem)35 b(6.1.)42 b Fl(L)-5 b(et)34 b Ft(n)26 b Fr(\025)g Fv(2)p Fl(.)44 b(F)-7 b(or)34 b(every)f Ft(n)p Fl(-torsion)h(p)-5 b(oint)35 b Ft(a)26 b Fr(2)g Ft(A)2446 3705 y Fq(n)2526 3691 y Fl(the)34 b(skyscr)-5 b(ap)g(er)35 b(she)-5 b(af)35 b Fs(C)p Fv(\()p Ft(\016)s Fv(\()p Ft(a)p Fv(\)\))g Fl(is)75 3804 y(a)e Fs(P)225 3771 y Fq(n)272 3804 y Fl(-obje)-5 b(ct)33 b(in)f Fv(D)744 3768 y Fq(b)744 3829 y Fn(S)799 3837 y Fz(n)845 3804 y Fv(\()p Ft(N)953 3818 y Fq(n)p Fo(\000)p Fp(1)1091 3804 y Ft(A)p Fv(\))p Fl(.)75 3989 y(Pr)-5 b(o)g(of.)43 b Fv(Indeed,)30 b(using)e(the)j(results)e(for)h (the)h(in)m(v)-5 b(arian)m(ts)29 b(of)i Fr(^)2238 3956 y Fo(\003)2277 3989 y Ft(N)2350 4007 y Fp(\001)p Fq(=)l(A)2493 3988 y Fz(n)2570 3989 y Fv(of)f(section)2977 3989 y SDict begin H.S end 2977 3989 a Fv(3)3022 3930 y SDict begin H.R end 3022 3930 a 3022 3989 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.3) cvn H.B /ANN pdfmark end 3022 3989 a 203 4194 a Fv(Hom)393 4156 y Fo(\003)393 4230 y Fp(D)447 4205 y Fz(b)447 4252 y Fi(S)495 4260 y Fz(n)545 4194 y Fv(\()p Fs(C)p Fv(\()p Ft(\016)s Fv(\()p Ft(a)p Fv(\)\))p Ft(;)15 b Fs(C)p Fv(\()p Ft(\016)s Fv(\()p Ft(a)p Fv(\)\)\))1298 4169 y Fr(\030)1298 4198 y Fv(=)1394 4194 y(Ext)1539 4156 y Fo(\003)1578 4194 y Fv(\()p Fs(C)p Fv(\()p Ft(\016)s Fv(\()p Ft(a)p Fv(\)\))p Ft(;)g Fs(C)p Fv(\()p Ft(\016)s Fv(\()p Ft(a)p Fv(\)\)\))2300 4156 y Fn(S)2355 4164 y Fz(n)2433 4169 y Fr(\030)2433 4198 y Fv(=)2529 4194 y Fr(^)2590 4156 y Fo(\003)2629 4194 y Ft(T)2682 4208 y Fq(N)2738 4217 y Fz(n)p Fy(\000)p Fx(1)2859 4208 y Fq(A)2916 4194 y Fv(\()p Ft(\016)s Fv(\()p Ft(a)p Fv(\)\))3147 4156 y Fn(S)3202 4164 y Fz(n)2433 4354 y Fr(\030)2433 4383 y Fv(=)2529 4379 y Fr(^)2590 4341 y Fo(\003)2629 4379 y Ft(N)2702 4397 y Fp(\001)p Fq(=)l(A)2845 4378 y Fz(n)2892 4379 y Fv(\()p Ft(\016)s Fv(\()p Ft(a)p Fv(\)\))3123 4341 y Fn(S)3178 4349 y Fz(n)2433 4494 y Fr(\030)2433 4523 y Fv(=)2529 4519 y Fs(C)20 b Fr(\010)g Fs(C)p Fv([)p Fr(\000)p Fv(2])h Fr(\010)f(\001)15 b(\001)g(\001)21 b(\010)f Fs(C)p Fv([)p Fr(\000)p Fv(2)p Ft(n)p Fv(])15 b Ft(:)3679 4704 y Fj(\003)75 4889 y SDict begin H.S end 75 4889 a 75 4889 a SDict begin 13 H.A end 75 4889 a 75 4889 a SDict begin [/View [/XYZ H.V]/Dest (theorem.6.2) cvn /DEST pdfmark end 75 4889 a FF(Remark)30 b(6.2.)39 b Fv(F)-8 b(or)27 b(t)m(w)m(o)h(di\013eren)m(t)e Ft(n)p Fv(-torsion)g(p)s(oin)m(ts)g(the)h(skyscrap)s(er)e(shea)m(v)m (es)j(are)f(orthogonal)g(whic)m(h)75 4997 y(mak)m(es)k(the)g(asso)s (ciated)g(t)m(wists)f(comm)m(ute.)42 b(Th)m(us,)29 b(w)m(e)i(ha)m(v)m (e)g(an)g(inclusion)c(\(see)k(corollary)3289 4997 y SDict begin H.S end 3289 4997 a Fv(2.4)3405 4939 y SDict begin H.R end 3405 4939 a 3405 4997 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.4) cvn H.B /ANN pdfmark end 3405 4997 a Fv(\))984 5199 y Fs(Z)1049 5162 y Fq(n)1092 5138 y Fx(4)1156 5199 y Fr(\032)25 b Fv(Aut\(D)1511 5162 y Fq(b)1511 5222 y Fn(S)1566 5230 y Fz(n)1613 5199 y Fv(\()p Ft(N)1721 5213 y Fq(n)p Fo(\000)p Fp(1)1858 5199 y Ft(A)p Fv(\)\))2022 5174 y Fr(\030)2022 5203 y Fv(=)2118 5199 y(Aut\(D)2377 5162 y Fq(b)2412 5199 y Fv(\()p Ft(K)2524 5213 y Fq(n)p Fo(\000)p Fp(1)2661 5199 y Ft(A)p Fv(\)\))15 b Ft(:)p eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 67 -8 a SDict begin H.S end 67 -8 a 67 -8 a SDict begin H.R end 67 -8 a 67 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.13) cvn /DEST pdfmark end 67 -8 a 329 66 a Fp(NEW)30 b(DERIVED)h(A)n(UTOEQUIV)-8 b(ALENCES)30 b(OF)h(HILBER)-6 b(T)31 b(SCHEMES)f(AND)g(KUMMER)g(V)-8 b(ARIETIES)183 b(13)175 266 y Fv(In)38 b(the)h(case)i Ft(n)e Fv(=)h(2)f(the)h (generalised)e(Kummer)g(v)-5 b(ariet)m(y)39 b Ft(K)2367 280 y Fq(n)p Fo(\000)p Fp(1)2505 266 y Ft(A)h Fv(=)f Ft(K)2800 280 y Fp(1)2840 266 y Ft(A)g Fv(is)f(just)h(the)g(Kummer)75 374 y(surface)30 b Ft(K)7 b Fv(\()p Ft(A)p Fv(\).)42 b(Moreo)m(v)m(er,)32 b(there)f(is)e(an)i(isomorphism)c(of)k(comm)m (utativ)m(e)h(diagrams)p 985 468 163 4 v 985 541 a Ft(I)1032 514 y Fq(n)1079 541 y Ft(A)1311 484 y Fq(p)1204 541 y Fr(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)126 b Ft(N)1652 555 y Fp(1)1691 541 y Ft(A)1002 702 y Fq(q)1036 612 y Fk(?)1036 667 y(?)1036 721 y(y)1639 612 y(?)1639 667 y(?)1639 721 y(y)1699 709 y Fq(\031)974 892 y Ft(K)1051 906 y Fp(1)1091 892 y Ft(A)45 b Fr(\000)-42 b(\000)-21 b(\000)h(\000)-43 b(!)1308 947 y Fq(\026)1499 892 y Ft(N)1572 906 y Fp(1)1611 892 y Ft(A=)p Fm(S)1799 906 y Fp(2)1956 705 y Fr(\030)1956 735 y Fv(=)2244 518 y(~)2220 541 y Ft(A)2517 484 y Fq(p)2411 541 y Fr(\000)g(\000)-20 b(\000)f(\000)-42 b(!)84 b Ft(A)2190 702 y Fq(q)2224 612 y Fk(?)2224 667 y(?)2224 721 y(y)2748 612 y(?)2748 667 y(?)2748 721 y(y)2808 709 y Fq(\031)2143 892 y Ft(K)7 b Fv(\()p Ft(A)p Fv(\))46 b Fr(\000)-43 b(\000)-20 b(\000)f(\000)-42 b(!)2514 947 y Fq(\026)2705 892 y Ft(A=\023)75 1082 y Fv(where)27 b Ft(p)h Fv(and)f Ft(\026)g Fv(in)g(the)h(righ)m(t-hand)e(diagram)h (are)h(the)g(blo)m(w-ups)f(of)g(the)h(16)h(di\013eren)m(t)e(2-torsion)h (p)s(oin)m(ts)75 1190 y(resp)s(ectiv)m(ely)h(of)g(their)g(image)g (under)f(the)i(quotien)m(t)f(under)f(the)i(in)m(v)m(olution)d Ft(\023)f Fv(=)f(\()p Fr(\000)p Fv(1\).)41 b(F)-8 b(or)30 b(a)g(2-torsion)75 1298 y(p)s(oin)m(t)f Ft(a)d Fr(2)e Ft(A)539 1312 y Fp(2)609 1298 y Fv(w)m(e)31 b(denote)g(b)m(y)f Ft(E)5 b Fv(\()p Ft(a)p Fv(\))31 b(the)f(exceptional)g(divisor)e(o)m(v) m(er)k(the)e(p)s(oin)m(t)g([)p Ft(a)p Fv(])25 b Fr(2)g Ft(A=\023)p Fv(.)42 b(Since)29 b Ft(E)5 b Fv(\()p Ft(a)p Fv(\))75 1406 y(is)35 b(a)h(rational)f(curv)m(e)i(in)d(the)i Ft(K)7 b Fv(3-surface)37 b Ft(K)7 b Fv(\()p Ft(A)p Fv(\),)38 b(ev)m(ery)e(line)e(bundle)g(on)i(it)f(is)g(a)h(spherical)e(ob)5 b(ject)37 b(in)75 1519 y(D)144 1483 y Fq(b)179 1519 y Fv(\()p Ft(K)7 b Fv(\()p Ft(A)p Fv(\)\).)75 1683 y SDict begin H.S end 75 1683 a 75 1683 a SDict begin 13 H.A end 75 1683 a 75 1683 a SDict begin [/View [/XYZ H.V]/Dest (theorem.6.3) cvn /DEST pdfmark end 75 1683 a FF(Prop)s(osition)36 b(6.3.)42 b Fl(F)-7 b(or)34 b(every)f(2-torsion)h(p)-5 b(oint)34 b Fv(\010)1940 1650 y Fo(\000)p Fp(1)2034 1683 y Fv(\()p Fs(C)p Fv(\()p Ft(\016)s Fv(\()p Ft(a)p Fv(\)\)\))28 b(=)d Fr(O)2606 1702 y Fq(E)t Fp(\()p Fq(a)p Fp(\))2758 1683 y Fv(\()p Fr(\000)p Fv(2\)[1])35 b Fl(holds.)75 1848 y(Pr)-5 b(o)g(of.)43 b Fv(Using)c(the)h(isomorphism)d(of)j(the)g (comm)m(utativ)m(e)h(diagrams)e(ab)s(o)m(v)m(e)i(the)f(pro)s(of)f(is)g (nearly)g(the)75 1956 y(same)31 b(as)f(the)h(pro)s(of)f(of)g(prop)s (osition)1390 1956 y SDict begin H.S end 1390 1956 a Fv(4.4)1506 1897 y SDict begin H.R end 1506 1897 a 1506 1956 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.4) cvn H.B /ANN pdfmark end 1506 1956 a Fv(.)2148 b Fj(\003)175 2119 y Fv(There)40 b(is)f(no)i(kno)m(wn)f(homomorphism)e(Aut\(D)1895 2083 y Fq(b)1930 2119 y Fv(\()p Ft(A)p Fv(\)\))43 b Fr(!)f Fv(Aut\(D)2538 2083 y Fq(b)2573 2119 y Fv(\()p Ft(K)2685 2133 y Fq(n)2732 2119 y Ft(A)p Fv(\)\))g(analogous)f(to)g(Plo)s(og's)75 2227 y(map)35 b Ft(\013)p Fv(.)55 b(But)36 b(at)f(least)h(one)f(can)h (lift)d(line)h(bundles)f Ft(L)g Fr(2)f Fv(Pic)15 b Ft(A)35 b Fv(\(b)m(y)h(restricting)e Fr(D)3057 2241 y Fq(L)3109 2227 y Fv(\))i(and)e(group)h(au-)75 2341 y(tomorphisms)c Ft(')d Fr(2)g Fv(Aut\()p Ft(A)p Fv(\))33 b(\(b)m(y)g(restricting)e Ft(')1780 2308 y Fp([)p Fq(n)p Fp(])1867 2341 y Fv(\))h(to)h(the)f (generalised)g(Kummer)f(v)-5 b(ariet)m(y)d(.)47 b(Recen)m(tly)-8 b(,)75 2449 y(Meac)m(han)30 b(has)f(sho)m(wn)g(in)e([)1022 2449 y SDict begin H.S end 1022 2449 a Fv(Mea12)1283 2387 y SDict begin H.R end 1283 2387 a 1283 2449 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Mea) cvn H.B /ANN pdfmark end 1283 2449 a Fv(])i(that)g(the)h(restriction)e(of)h(Addington's)e (functor)i(to)h(the)f(generalised)75 2557 y(Kummer)d(v)-5 b(ariet)m(y)27 b Ft(K)825 2571 y Fq(n)873 2557 y Fv(\()p Ft(A)p Fv(\))h(for)e Ft(n)f Fr(\025)g Fv(2)j(\(i.e.)f(the)h(F)-8 b(ourier{Muk)j(ai)26 b(transform)h(with)f(k)m(ernel)g(the)i(univ)m (ersal)75 2665 y(sheaf)7 b(\))32 b(is)f(still)e(a)j Fs(P)768 2632 y Fq(n)p Fo(\000)p Fp(1)937 2665 y Fv(functor)f(and)g(th)m(us)g (yields)f(an)h(auto)s(equiv)-5 b(alence)33 b(\026)-46 b Ft(a)p Fv(.)44 b(Comparing)30 b(these)i(auto)s(e-)75 2772 y(quiv)-5 b(alences)26 b(with)f(those)i(induced)e(b)m(y)h(the)h (ab)s(o)m(v)m(e)h Fs(P)1919 2739 y Fq(n)1966 2772 y Fv(-ob)5 b(jects)28 b(one)f(gets)g(results)f(similar)e(to)j(the)g(results)75 2880 y(of)k(section)482 2880 y SDict begin H.S end 482 2880 a Fv(5)528 2822 y SDict begin H.R end 528 2822 a 528 2880 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.5) cvn H.B /ANN pdfmark end 528 2880 a Fv(.)1647 3076 y SDict begin H.S end 1647 3076 a 1647 3076 a SDict begin 13 H.A end 1647 3076 a 1647 3076 a SDict begin [/View [/XYZ H.V]/Dest (section*.1) cvn /DEST pdfmark end 1647 3076 a Fu(References)75 3221 y SDict begin H.S end 75 3221 a FE([A)n(C12])304 3221 y SDict begin 11 H.A end 304 3221 a 304 3221 a SDict begin [/View [/XYZ H.V]/Dest (cite.AC) cvn /DEST pdfmark end 304 3221 a 99 w FE(Dima)25 b(Arinkin)g(and)g(Andrei)g(C\025)-38 b(ald\025)g(araru.)27 b(When)e(is)h(the)f(self-in)n(tersection)i(of)f(a)g(sub)n(v)l(ariet)n (y)f(a)h(\014bration?)34 b Fc(A)l(dv.)403 3313 y(Math.)p FE(,)27 b(231\(2\):815{842,)j(2012.)75 3404 y SDict begin H.S end 75 3404 a FE([Add11])337 3404 y SDict begin 11 H.A end 337 3404 a 337 3404 a SDict begin [/View [/XYZ H.V]/Dest (cite.Add) cvn /DEST pdfmark end 337 3404 a 66 w FE(Nicolas)e(Addington.)d(New)h(deriv)n(ed)f(symmetries)f(of)i (some)g(h)n(yp)r(erk)l(aehler)e(v)l(arieties.)j Fc(arXiv:1112.0487)p FE(,)g(2011.)75 3495 y SDict begin H.S end 75 3495 a FE([AL12])299 3495 y SDict begin 11 H.A end 299 3495 a 299 3495 a SDict begin [/View [/XYZ H.V]/Dest (cite.AL) cvn /DEST pdfmark end 299 3495 a 104 w FE(Rina)34 b(Anno)e(and)h (Timoth)n(y)f(Logvinenk)n(o.)i(On)e(adjunctions)i(for)g(Fourier{Muk)l (ai)g(transforms.)g Fc(A)l(dv.)g(Math.)p FE(,)403 3587 y(231\(3-4\):2069{2115,)e(2012.)75 3678 y SDict begin H.S end 75 3678 a FE([Ann07])337 3678 y SDict begin 11 H.A end 337 3678 a 337 3678 a SDict begin [/View [/XYZ H.V]/Dest (cite.Ann) cvn /DEST pdfmark end 337 3678 a 66 w FE(Rina)26 b(Anno.)f(Spherical)h(functors.)h Fc(arXiv:0711.4409)p FE(,)g(2007.)75 3769 y SDict begin H.S end 75 3769 a FE([BKR01])364 3769 y SDict begin 11 H.A end 364 3769 a 364 3769 a SDict begin [/View [/XYZ H.V]/Dest (cite.BKR) cvn /DEST pdfmark end 364 3769 a 39 w FE(T)-6 b(om)31 b(Bridgeland,)i(Alastair)g(King,)f(and)f(Miles)h(Reid.)f(The)g(McKa)n (y)g(corresp)r(ondence)h(as)f(an)g(equiv)l(alence)g(of)403 3861 y(deriv)n(ed)26 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b(olge.)31 b(A)g(Series)i(of)e(Mo)l(dern)h(Surveys)h(in)f(Mathematics)g ([R)l(esults)h(in)e(Mathematics)i(and)f(R)l(elate)l(d)g(A)n(r)l(e)l (as.)403 4317 y(3r)l(d)j(Series.)g(A)f(Series)h(of)e(Mo)l(dern)i (Surveys)h(in)e(Mathematics])p FE(.)g(Springer-V)-6 b(erlag,)35 b(Berlin,)g(second)e(edition,)403 4408 y(1998.)75 4500 y SDict begin H.S end 75 4500 a FE([Hai01])310 4500 y SDict begin 11 H.A end 310 4500 a 310 4500 a SDict begin [/View [/XYZ H.V]/Dest (cite.Hai) cvn /DEST pdfmark end 310 4500 a 93 w FE(Mark)20 b(Haiman.)f(Hilb)r(ert)h(sc)n(hemes,)g(p)r (olygraphs)g(and)f(the)g(Macdonald)i(p)r(ositivit)n(y)e(conjecture.)i Fc(J.)g(A)n(mer.)h(Math.)403 4591 y(So)l(c.)p FE(,)27 b(14\(4\):941{1006)j(\(electronic\),)d(2001.)75 4682 y SDict begin H.S end 75 4682 a FE([Har66])319 4682 y SDict begin 11 H.A end 319 4682 a 319 4682 a SDict begin [/View [/XYZ H.V]/Dest (cite.Har1) cvn /DEST pdfmark end 319 4682 a 84 w FE(Robin)c(Hartshorne.)f Fc(R)l(esidues)k(and)f (duality)p 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