Algebraic Geometry

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aria-labelledby="recent-math.AG" href="/list/math.AG/recent">recent</a> articles</p> <h3>Showing new listings for Tuesday, 26 November 2024</h3> <div class='paging'>Total of 43 entries </div> <div class='morefewer'>Showing up to 2000 entries per page: <a href=/list/math.AG/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> <dl id='articles'> <h3>New submissions (showing 14 of 14 entries)</h3> <dt> <a name='item1'>[1]</a> <a href ="/abs/2411.15334" title="Abstract" id="2411.15334"> arXiv:2411.15334 </a> [<a href="/pdf/2411.15334" title="Download PDF" id="pdf-2411.15334" aria-labelledby="pdf-2411.15334">pdf</a>, <a href="https://arxiv.org/html/2411.15334v1" title="View HTML" id="html-2411.15334" aria-labelledby="html-2411.15334" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.15334" title="Other formats" id="oth-2411.15334" aria-labelledby="oth-2411.15334">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Icosahedron in birational geometry </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Prokhorov,+Y">Yuri Prokhorov</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 28 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> We study quotients of projective and affine spaces by various actions of the icosahedral group. Basically we concentrate on the rationality questions. </p> </div> </dd> <dt> <a name='item2'>[2]</a> <a href ="/abs/2411.15353" title="Abstract" id="2411.15353"> arXiv:2411.15353 </a> [<a href="/pdf/2411.15353" title="Download PDF" id="pdf-2411.15353" aria-labelledby="pdf-2411.15353">pdf</a>, <a href="https://arxiv.org/html/2411.15353v1" title="View HTML" id="html-2411.15353" aria-labelledby="html-2411.15353" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.15353" title="Other formats" id="oth-2411.15353" aria-labelledby="oth-2411.15353">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On spectral sequences for semiabelian varieties over non-closed fields </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Petrov,+A">Alexander Petrov</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Skorobogatov,+A">Alexei Skorobogatov</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 34 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> We give a new, short proof of the formula for the first potentially non-zero differential of the Hochschild--Serre spectral sequence for semiabelian varieties over non-closed fields. We show that this differential is non-zero for the Jacobian of a curve when the image of the torsor of theta-characteristics under the Bockstein map is non-zero. An explicit example is a curve of genus 2 whose Albanese torsor is not divisible by 2. When the Albanese torsor is trivial, we show that the Hochschild--Serre spectral sequence for the Jacobian degenerates at the second page. We give a formula for the differential of the Hochschild--Serre spectral sequence for a torus which computes its Brauer group. Finally, we describe the differentials of the Hochschild--Serre spectral sequence for a smooth projective curve, generalising a lemma of Suslin. </p> </div> </dd> <dt> <a name='item3'>[3]</a> <a href ="/abs/2411.15389" title="Abstract" id="2411.15389"> arXiv:2411.15389 </a> [<a href="/pdf/2411.15389" title="Download PDF" id="pdf-2411.15389" aria-labelledby="pdf-2411.15389">pdf</a>, <a href="https://arxiv.org/html/2411.15389v1" title="View HTML" id="html-2411.15389" aria-labelledby="html-2411.15389" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.15389" title="Other formats" id="oth-2411.15389" aria-labelledby="oth-2411.15389">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Singularity categories and singular loci of certain quotient singularities </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Chen,+X">Xiaojun Chen</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Zeng,+J">Jieheng Zeng</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Rings and Algebras (math.RA); Representation Theory (math.RT) </div> <p class='mathjax'> Let $V$ be a finite dimensional $k$-vector space, where $k$ is an algebraic closed field of characteristic zero. Let $G \subseteq\ mathrm{SL}(V)$ be a finite abelian group, and denote by $S$ the $G$-invariant subring of the polynomial ring $k[V]$. It is shown that the singularity category $D_{sg}(S)$ recovers the reduced singular locus of $\mathrm{Spec}(S)$. </p> </div> </dd> <dt> <a name='item4'>[4]</a> <a href ="/abs/2411.15606" title="Abstract" id="2411.15606"> arXiv:2411.15606 </a> [<a href="/pdf/2411.15606" title="Download PDF" id="pdf-2411.15606" aria-labelledby="pdf-2411.15606">pdf</a>, <a href="https://arxiv.org/html/2411.15606v1" title="View HTML" id="html-2411.15606" aria-labelledby="html-2411.15606" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.15606" title="Other formats" id="oth-2411.15606" aria-labelledby="oth-2411.15606">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> A polyptych of multi-centered deformation spaces </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Dubouloz,+A">Adrien Dubouloz</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Mayeux,+A">Arnaud Mayeux</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> We study deformation spaces using multi-centered dilatations. Interpolating Fulton simple deformation space and Rost asymmetric double deformation space, we introduce (asymmetric) deformation spaces attached to chains of immersions of arbitrary lengths. One of the main results of this paper is the so-called panelization isomorphism, producing several isomorphisms between the deformation space of length $n$ and deformation spaces of smaller lengths. Combining these isomorphisms, we get a polyptych $\mathscr{P}(n)$ of deformation spaces. Having these panelization isomorphisms allows to compute the strata -- certain restrictions of special interests -- of deformation spaces. </p> </div> </dd> <dt> <a name='item5'>[5]</a> <a href ="/abs/2411.15774" title="Abstract" id="2411.15774"> arXiv:2411.15774 </a> [<a href="/pdf/2411.15774" title="Download PDF" id="pdf-2411.15774" aria-labelledby="pdf-2411.15774">pdf</a>, <a href="https://arxiv.org/html/2411.15774v1" title="View HTML" id="html-2411.15774" aria-labelledby="html-2411.15774" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.15774" title="Other formats" id="oth-2411.15774" aria-labelledby="oth-2411.15774">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> A remark on a theorem of Narasimhan and Ramanan </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Pine,+J">Jagadish Pine</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> In this short note, we provide an alternative proof of a notable theorem by Narasimhan and Ramanan. The theorem states that the moduli space of $S$-equivalence classes of semistable rank $2$ vector bundles over a curve $X$ of genus $2$ with trivial determinant is isomorphic to $\mathbb{P}^3$. Our proof relies on a criterion by Bauer and Szemberg, which characterizes projective spaces among smooth Fano varieties using Seshadri constants. </p> </div> </dd> <dt> <a name='item6'>[6]</a> <a href ="/abs/2411.15879" title="Abstract" id="2411.15879"> arXiv:2411.15879 </a> [<a href="/pdf/2411.15879" title="Download PDF" id="pdf-2411.15879" aria-labelledby="pdf-2411.15879">pdf</a>, <a href="https://arxiv.org/html/2411.15879v1" title="View HTML" id="html-2411.15879" aria-labelledby="html-2411.15879" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.15879" title="Other formats" id="oth-2411.15879" aria-labelledby="oth-2411.15879">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> A note on smooth $SL_2$-surfaces </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Freudenburg,+G">Gene Freudenburg</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> Working over a field $k$ of characteristic zero, we study the ring $\mathfrak{R}=\mathfrak{D}^{\mathbb{Z}_2}$ where $\mathfrak{D}=k[x_0,x_1,x_2]/(2x_0x_2-x_1^2-1)$ and $\mathbb{Z}_2$ acts by $x_i\to -x_i$. $\mathfrak{D}$ admits an algebraic $SL_2(k)$-action which restricts to $\mathfrak{R}$. Our results include the following. (1) If $k$ is algebraically closed, the smooth $SL_2$-surface $X={\rm Spec}(\mathfrak{R})$ admits an algebraic embedding in $\mathbb{A}_k^4$, and for any such embedding the $SL_2(k)$-action on $X$ does not extend to $\mathbb{A}_k^4$. In addition, there is no algebraic embedding of $X$ in $\mathbb{A}_k^3$. (2) The automorphism group ${\rm Aut}_k(\mathfrak{R})$ acts transitively on the set of irreducible locally nilpotent derivations of $\mathfrak{R}$. (3) Every automorphism of $\mathfrak{R}$ extends to $\mathfrak{D}$, and ${\rm Aut}_k(\mathfrak{R})=PSL_2(k)\ast_HT$ where $T$ is its triangular subgroup. (4) $\mathfrak{R}$ is non-cancellative, i.e., there exists a ring $\mathfrak{S}$ such that $\mathfrak{R}^{[1]}\cong_k\mathfrak{S}^{[1]}$ but $\mathfrak{R}\not\cong_k\mathfrak{S}$. In order to distinguish $\mathfrak{R}$ from $\mathfrak{S}$, we calculate the plinth invariant for $\mathfrak{R}$. </p> </div> </dd> <dt> <a name='item7'>[7]</a> <a href ="/abs/2411.15905" title="Abstract" id="2411.15905"> arXiv:2411.15905 </a> [<a href="/pdf/2411.15905" title="Download PDF" id="pdf-2411.15905" aria-labelledby="pdf-2411.15905">pdf</a>, <a href="/format/2411.15905" title="Other formats" id="oth-2411.15905" aria-labelledby="oth-2411.15905">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Diagonalization of Operator functions by algebraic methods </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Stiefenhofer,+M">Matthias Stiefenhofer</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 27 pages. arXiv admin note: text overlap with <a href="https://arxiv.org/abs/2305.14228" data-arxiv-id="2305.14228" class="link-https">arXiv:2305.14228</a> </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> We give conditions for local diagonalization of an analytic operator family to a diagonal operator polynomial. The families are acting between real or complex Banach spaces. The basic assumption is given by stabilization of the Jordan chains at length k in the sense that no root elements with finite rank above k are allowed to exist. Jordan chains with infinite rank may appear. Decompositions of the linear spaces are constructed with corresponding subspaces assumed to be closed. These assumptions ensure finite pole order equal to k of the generalized inverse. The Smith form and smooth continuation of kernels and ranges to appropriate limit spaces arise immediately. An algebraically oriented and self-contained approach is used, based on a recursion that allows for construction of power series solutions. The power series solutions are convergent, as soon as analyticity and continuity of related projections are assumed. </p> </div> </dd> <dt> <a name='item8'>[8]</a> <a href ="/abs/2411.15935" title="Abstract" id="2411.15935"> arXiv:2411.15935 </a> [<a href="/pdf/2411.15935" title="Download PDF" id="pdf-2411.15935" aria-labelledby="pdf-2411.15935">pdf</a>, <a href="https://arxiv.org/html/2411.15935v1" title="View HTML" id="html-2411.15935" aria-labelledby="html-2411.15935" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.15935" title="Other formats" id="oth-2411.15935" aria-labelledby="oth-2411.15935">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Decomposing tensors via rank-one approximations </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Ribot,+A">Alvaro Ribot</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Horobet,+E">Emil Horobet</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Seigal,+A">Anna Seigal</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Turatti,+E+T">Ettore Teixeira Turatti</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 26 pages, 1 figure </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Spectral Theory (math.SP) </div> <p class='mathjax'> Matrices can be decomposed via rank-one approximations: the best rank-one approximation is a singular vector pair, and the singular value decomposition writes a matrix as a sum of singular vector pairs. The singular vector tuples of a tensor are the critical points of its best rank-one approximation problem. In this paper, we study tensors that can be decomposed via successive rank-one approximations: compute a singular vector tuple, subtract it off, compute a singular vector tuple of the new deflated tensor, and repeat. The number of terms in such a decomposition may exceed the tensor rank. Moreover, the decomposition may depend on the order in which terms are subtracted. We show that the decomposition is valid independent of order if and only if all singular vectors in the process are orthogonal in at least two factors. We study the variety of such tensors. We lower bound its dimension, showing that it is significantly larger than the variety of odeco tensors. </p> </div> </dd> <dt> <a name='item9'>[9]</a> <a href ="/abs/2411.16041" title="Abstract" id="2411.16041"> arXiv:2411.16041 </a> [<a href="/pdf/2411.16041" title="Download PDF" id="pdf-2411.16041" aria-labelledby="pdf-2411.16041">pdf</a>, <a href="https://arxiv.org/html/2411.16041v1" title="View HTML" id="html-2411.16041" aria-labelledby="html-2411.16041" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.16041" title="Other formats" id="oth-2411.16041" aria-labelledby="oth-2411.16041">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Witt Group of Nondyadic Curves </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Yang,+N">Nanjun Yang</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 15pages;comments are welcomed! </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; K-Theory and Homology (math.KT); Number Theory (math.NT) </div> <p class='mathjax'> We compute the Witt group of smooth proper curves over nondyadic local fields with $char\neq2$ by reduction. </p> </div> </dd> <dt> <a name='item10'>[10]</a> <a href ="/abs/2411.16141" title="Abstract" id="2411.16141"> arXiv:2411.16141 </a> [<a href="/pdf/2411.16141" title="Download PDF" id="pdf-2411.16141" aria-labelledby="pdf-2411.16141">pdf</a>, <a href="https://arxiv.org/html/2411.16141v1" title="View HTML" id="html-2411.16141" aria-labelledby="html-2411.16141" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.16141" title="Other formats" id="oth-2411.16141" aria-labelledby="oth-2411.16141">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Stable maps to quotient stacks with a properly stable point </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Di+Lorenzo,+A">Andrea Di Lorenzo</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Inchiostro,+G">Giovanni Inchiostro</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> Comments are welcome! </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> We compactify the moduli stack of maps from curves to certain quotient stacks $X = [W/G]$ with a projective good moduli space, extending previous results from quasimap theory. For doing so, we introduce a new birational transformation for algebraic stacks, the extended weighted blow-up, to prove that any algebraic stack with a properly stable point can be enlarged so that it contains an open substack which is proper and Deligne-Mumford. We give a criterion for when a morphism of algebraic stacks is an extended weighted blow-up, and we use it in order to give a modular proof of a conjecture of Hassett on weighted pointed rational curves. Finally, we present some applications of our main results when X is respectively a quotient by a torus of a proper Deligne-Mumford stack; a GIT compactification of the stack of binary forms of degree 2n; a GIT compactification of the stack of 2n-marked smooth rational curves, and a GIT compactification of the stack of smooth plane cubics. </p> </div> </dd> <dt> <a name='item11'>[11]</a> <a href ="/abs/2411.16146" title="Abstract" id="2411.16146"> arXiv:2411.16146 </a> [<a href="/pdf/2411.16146" title="Download PDF" id="pdf-2411.16146" aria-labelledby="pdf-2411.16146">pdf</a>, <a href="https://arxiv.org/html/2411.16146v1" title="View HTML" id="html-2411.16146" aria-labelledby="html-2411.16146" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.16146" title="Other formats" id="oth-2411.16146" aria-labelledby="oth-2411.16146">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On the divisorial contractions to curves of threefolds </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Chen,+H">Hsin-Ku Chen</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Chen,+J">Jheng-Jie Chen</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Chen,+J+A">Jungkai A. Chen</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> We prove that each divisorial contraction to a curve between terminal threefolds is a weighted blow-up under a suitable embedding. Moreover, we give a classification of the weighted blow-ups assuming that the curve is smooth. </p> </div> </dd> <dt> <a name='item12'>[12]</a> <a href ="/abs/2411.16540" title="Abstract" id="2411.16540"> arXiv:2411.16540 </a> [<a href="/pdf/2411.16540" title="Download PDF" id="pdf-2411.16540" aria-labelledby="pdf-2411.16540">pdf</a>, <a href="https://arxiv.org/html/2411.16540v1" title="View HTML" id="html-2411.16540" aria-labelledby="html-2411.16540" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.16540" title="Other formats" id="oth-2411.16540" aria-labelledby="oth-2411.16540">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Isotropic motivic fundamental groups </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Tanania,+F">Fabio Tanania</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Algebraic Topology (math.AT); K-Theory and Homology (math.KT) </div> <p class='mathjax'> The main goal of this paper is to study relative versions of the category of modules over the isotropic motivic Brown-Peterson spectrum, with a particular emphasis on their cellular subcategories. Using techniques developed by Levine, we equip these categories with motivic $t$-structures, whose hearts are Tannakian categories over ${\mathbb F}_2$. This allows to define isotropic motivic fundamental groups, and to interpret relative isotropic Tate motives in the heart as their representations. Moreover, we compute these groups in the cases of the punctured projective line and split tori. Finally, we also apply Spitzweck's derived approach to establish an identification between relative isotropic Tate motives and representations of certain affine derived group schemes, whose 0-truncations coincide with the aforementioned isotropic motivic fundamental groups. </p> </div> </dd> <dt> <a name='item13'>[13]</a> <a href ="/abs/2411.16664" title="Abstract" id="2411.16664"> arXiv:2411.16664 </a> [<a href="/pdf/2411.16664" title="Download PDF" id="pdf-2411.16664" aria-labelledby="pdf-2411.16664">pdf</a>, <a href="https://arxiv.org/html/2411.16664v1" title="View HTML" id="html-2411.16664" aria-labelledby="html-2411.16664" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.16664" title="Other formats" id="oth-2411.16664" aria-labelledby="oth-2411.16664">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Slope semistability of Veronese normal bundles </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Shang,+R">Ray Shang</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 17 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> A classical fact is that normal bundles of rational normal curves are well-balanced. We generalize this by proving that all Veronese normal bundles are slope semistable. We also determine the line bundle decomposition of the restriction of degree 2 Veronese normal bundles to lines and rational normal curves. </p> </div> </dd> <dt> <a name='item14'>[14]</a> <a href ="/abs/2411.16672" title="Abstract" id="2411.16672"> arXiv:2411.16672 </a> [<a href="/pdf/2411.16672" title="Download PDF" id="pdf-2411.16672" aria-labelledby="pdf-2411.16672">pdf</a>, <a href="https://arxiv.org/html/2411.16672v1" title="View HTML" id="html-2411.16672" aria-labelledby="html-2411.16672" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.16672" title="Other formats" id="oth-2411.16672" aria-labelledby="oth-2411.16672">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Interpolation for degree 2 Veroneses of odd dimension </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Shang,+R">Ray Shang</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 21 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> A classical fact is that through any $d+3$ general points in $\mathbb{P}_\mathbb{C}^d$ there exists a unique rational normal curve of degree $d$ passing through them. We generalize this by proving the following: when $n$ is odd, for any $\binom{n+2}{2} + n+1$ general points in $\mathbb{P}_\mathbb{C}^{\binom{n+2}{2} - 1}$, there exist at least $2^{n(n-1)}$ degree 2 Veroneses passing through them. This makes substantial progress on a question of Aaron Landesman and Anand Patel, and extends the work of Arthur Coble. </p> </div> </dd> </dl> <dl id='articles'> <h3>Cross submissions (showing 7 of 7 entries)</h3> <dt> <a name='item15'>[15]</a> <a href ="/abs/2411.15347" title="Abstract" id="2411.15347"> arXiv:2411.15347 </a> (cross-list from math.AT) [<a href="/pdf/2411.15347" title="Download PDF" id="pdf-2411.15347" aria-labelledby="pdf-2411.15347">pdf</a>, <a href="https://arxiv.org/html/2411.15347v1" title="View HTML" id="html-2411.15347" aria-labelledby="html-2411.15347" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.15347" title="Other formats" id="oth-2411.15347" aria-labelledby="oth-2411.15347">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Motivic configurations on the line </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Igieobo,+J">John Igieobo</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=McKean,+S">Stephen McKean</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Sanchez,+S">Steven Sanchez</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Taylor,+D">Dae'Shawn Taylor</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Wickelgren,+K">Kirsten Wickelgren</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 38 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Topology (math.AT)</span>; Algebraic Geometry (math.AG) </div> <p class='mathjax'> For each configuration of rational points on the affine line, we define an operation on the group of unstable A1 motivic homotopy classes of endomorphisms of the projective line. We also derive an algebraic formula for the image of such an operation under Cazanave and Morel's unstable degree map, which is valued in an extension of the Grothendieck--Witt group. In contrast to the topological setting, these operations depend on the choice of configuration of points via a discriminant. We prove this by first showing a local-to-global formula for the global unstable degree as a modified sum of local terms. We then use an anabelian argument to generalize from the case of local degrees of a global rational function to the case of an arbitrary collection of endomorphisms of the projective line. </p> </div> </dd> <dt> <a name='item16'>[16]</a> <a href ="/abs/2411.15789" title="Abstract" id="2411.15789"> arXiv:2411.15789 </a> (cross-list from cs.CC) [<a href="/pdf/2411.15789" title="Download PDF" id="pdf-2411.15789" aria-labelledby="pdf-2411.15789">pdf</a>, <a href="https://arxiv.org/html/2411.15789v1" title="View HTML" id="html-2411.15789" aria-labelledby="html-2411.15789" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.15789" title="Other formats" id="oth-2411.15789" aria-labelledby="oth-2411.15789">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Asymptotic tensor rank is characterized by polynomials </div> <div class='list-authors'><a href="https://arxiv.org/search/cs?searchtype=author&query=Christandl,+M">Matthias Christandl</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Hoeberechts,+K">Koen Hoeberechts</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Nieuwboer,+H">Harold Nieuwboer</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Vrana,+P">P茅ter Vrana</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Zuiddam,+J">Jeroen Zuiddam</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Computational Complexity (cs.CC)</span>; Algebraic Geometry (math.AG); Quantum Physics (quant-ph) </div> <p class='mathjax'> Asymptotic tensor rank is notoriously difficult to determine. Indeed, determining its value for the $2\times 2$ matrix multiplication tensor would determine the matrix multiplication exponent, a long-standing open problem. On the other hand, Strassen's asymptotic rank conjecture makes the bold claim that asymptotic tensor rank equals the largest dimension of the tensor and is thus as easy to compute as matrix rank. Despite tremendous interest, much is still unknown about the structural and computational properties of asymptotic rank; for instance whether it is computable. <br>We prove that asymptotic tensor rank is "computable from above", that is, for any real number $r$ there is an (efficient) algorithm that determines, given a tensor $T$, if the asymptotic tensor rank of $T$ is at most $r$. The algorithm has a simple structure; it consists of evaluating a finite list of polynomials on the tensor. Indeed, we prove that the sublevel sets of asymptotic rank are Zariski-closed (just like matrix rank). While we do not exhibit these polynomials explicitly, their mere existence has strong implications on the structure of asymptotic rank. <br>As one such implication, we find that the values that asymptotic tensor rank takes, on all tensors, is a well-ordered set. In other words, any non-increasing sequence of asymptotic ranks stabilizes ("discreteness from above"). In particular, for the matrix multiplication exponent (which is an asymptotic rank) there is no sequence of exponents of bilinear maps that approximates it arbitrarily closely from above without being eventually constant. In other words, any upper bound on the matrix multiplication exponent that is close enough, will "snap" to it. Previously such discreteness results were only known for finite fields or for other tensor parameters (e.g., asymptotic slice rank). We obtain them for infinite fields like the complex numbers. </p> </div> </dd> <dt> <a name='item17'>[17]</a> <a href ="/abs/2411.16033" title="Abstract" id="2411.16033"> arXiv:2411.16033 </a> (cross-list from hep-th) [<a href="/pdf/2411.16033" title="Download PDF" id="pdf-2411.16033" aria-labelledby="pdf-2411.16033">pdf</a>, <a href="https://arxiv.org/html/2411.16033v1" title="View HTML" id="html-2411.16033" aria-labelledby="html-2411.16033" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.16033" title="Other formats" id="oth-2411.16033" aria-labelledby="oth-2411.16033">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Generative AI for Brane Configurations, Tropical Coamoeba and 4d N=1 Quiver Gauge Theories </div> <div class='list-authors'><a href="https://arxiv.org/search/hep-th?searchtype=author&query=Seong,+R">Rak-Kyeong Seong</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 21 pages, 8 figures, 1 table </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">High Energy Physics - Theory (hep-th)</span>; Machine Learning (cs.LG); Mathematical Physics (math-ph); Algebraic Geometry (math.AG) </div> <p class='mathjax'> We introduce a generative AI model to obtain Type IIB brane configurations that realize toric phases of a family of 4d N=1 supersymmetric gauge theories. These 4d N=1 quiver gauge theories are worldvolume theories of a D3-brane probing a toric Calabi-Yau 3-fold. The Type IIB brane configurations that realize this family of 4d N=1 theories are known as brane tilings and are given by the tropical coamoeba projection of the mirror curve associated with the toric Calabi-Yau 3-fold. The shape of the mirror curve and its coamoeba projection, as well as the corresponding Type IIB brane configuration and the toric phase of the 4d N=1 theory, all depend on the complex structure moduli parameterizing the mirror curve. We train a generative AI model, a conditional variational autoencoder (CVAE), that takes a choice of complex structure moduli as input and generates the corresponding tropical coamoeba. This enables us not only to obtain a high-resolution representation of the entire phase space for a family of brane tilings corresponding to the same toric Calabi-Yau 3-fold, but also to continuously track the movements of the mirror curve and individual branes in the corresponding Type IIB brane configurations during phase transitions associated with Seiberg duality. </p> </div> </dd> <dt> <a name='item18'>[18]</a> <a href ="/abs/2411.16108" title="Abstract" id="2411.16108"> arXiv:2411.16108 </a> (cross-list from math.NT) [<a href="/pdf/2411.16108" title="Download PDF" id="pdf-2411.16108" aria-labelledby="pdf-2411.16108">pdf</a>, <a href="https://arxiv.org/html/2411.16108v1" title="View HTML" id="html-2411.16108" aria-labelledby="html-2411.16108" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.16108" title="Other formats" id="oth-2411.16108" aria-labelledby="oth-2411.16108">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Intersecting subvarieties of abelian schemes with group subschemes I </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Ge,+T">Tangli Ge</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 42 pages; comments are welcome! </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Number Theory (math.NT)</span>; Algebraic Geometry (math.AG) </div> <p class='mathjax'> In this paper, we establish the following family version of Habegger's bounded height theorem on abelian varieties: a locally closed subvariety of an abelian scheme with Gao's $t^{\mathrm{th}}$ degeneracy locus removed, intersected with all flat group subschemes of relative dimension at most $t$, gives a set of bounded total height. Our main tools include the Ax--Schanuel theorem, and intersection theory of adelic line bundles as developed by Yuan--Zhang. As two applications, we generalize Silverman's specialization theorem to a higher dimensional base, and establish a bounded height result towards Zhang's ICM Conjecture. </p> </div> </dd> <dt> <a name='item19'>[19]</a> <a href ="/abs/2411.16396" title="Abstract" id="2411.16396"> arXiv:2411.16396 </a> (cross-list from quant-ph) [<a href="/pdf/2411.16396" title="Download PDF" id="pdf-2411.16396" aria-labelledby="pdf-2411.16396">pdf</a>, <a href="https://arxiv.org/html/2411.16396v1" title="View HTML" id="html-2411.16396" aria-labelledby="html-2411.16396" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.16396" title="Other formats" id="oth-2411.16396" aria-labelledby="oth-2411.16396">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Statistical inference for quantum singular models </div> <div class='list-authors'><a href="https://arxiv.org/search/quant-ph?searchtype=author&query=Yano,+H">Hiroshi Yano</a>, <a href="https://arxiv.org/search/quant-ph?searchtype=author&query=Maeda,+Y">Yota Maeda</a>, <a href="https://arxiv.org/search/quant-ph?searchtype=author&query=Yamamoto,+N">Naoki Yamamoto</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 57 pages, 8 figures </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Quantum Physics (quant-ph)</span>; Machine Learning (cs.LG); Algebraic Geometry (math.AG); Machine Learning (stat.ML) </div> <p class='mathjax'> Deep learning has seen substantial achievements, with numerical and theoretical evidence suggesting that singularities of statistical models are considered a contributing factor to its performance. From this remarkable success of classical statistical models, it is naturally expected that quantum singular models will play a vital role in many quantum statistical tasks. However, while the theory of quantum statistical models in regular cases has been established, theoretical understanding of quantum singular models is still limited. To investigate the statistical properties of quantum singular models, we focus on two prominent tasks in quantum statistical inference: quantum state estimation and model selection. In particular, we base our study on classical singular learning theory and seek to extend it within the framework of Bayesian quantum state estimation. To this end, we define quantum generalization and training loss functions and give their asymptotic expansions through algebraic geometrical methods. The key idea of the proof is the introduction of a quantum analog of the likelihood function using classical shadows. Consequently, we construct an asymptotically unbiased estimator of the quantum generalization loss, the quantum widely applicable information criterion (QWAIC), as a computable model selection metric from given measurement outcomes. </p> </div> </dd> <dt> <a name='item20'>[20]</a> <a href ="/abs/2411.16543" title="Abstract" id="2411.16543"> arXiv:2411.16543 </a> (cross-list from math.SG) [<a href="/pdf/2411.16543" title="Download PDF" id="pdf-2411.16543" aria-labelledby="pdf-2411.16543">pdf</a>, <a href="https://arxiv.org/html/2411.16543v1" title="View HTML" id="html-2411.16543" aria-labelledby="html-2411.16543" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.16543" title="Other formats" id="oth-2411.16543" aria-labelledby="oth-2411.16543">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Fourier transforms and a filtration on the Lagrangian cobordism group of tori </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Mu%C3%B1iz-Brea,+%C3%81">脕lvaro Mu帽iz-Brea</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 36 pages, 3 figures </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Symplectic Geometry (math.SG)</span>; Algebraic Geometry (math.AG) </div> <p class='mathjax'> Given a polarized tropical affine torus, we show that the fibered Lagrangian cobordism group of the corresponding symplectic manifold admits a natural geometric filtration of finite length. This contrasts with results of Sheridan-Smith in dimension four and the present author in higher dimensions, who showed that such group is infinite-dimensional. <br>In the second half of this paper, we construct a Fourier transform between Fukaya categories of dual symplectic tori. We show that, under homological mirror symmetry, it corresponds to the Fourier transform between derived categories of coherent sheaves of dual abelian varieties due to Mukai. We use this to show how our filtration is mirror to the Bloch filtration on Chow groups of abelian varieties, but the results may be of broader interest. </p> </div> </dd> <dt> <a name='item21'>[21]</a> <a href ="/abs/2411.16589" title="Abstract" id="2411.16589"> arXiv:2411.16589 </a> (cross-list from math.DG) [<a href="/pdf/2411.16589" title="Download PDF" id="pdf-2411.16589" aria-labelledby="pdf-2411.16589">pdf</a>, <a href="https://arxiv.org/html/2411.16589v1" title="View HTML" id="html-2411.16589" aria-labelledby="html-2411.16589" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.16589" title="Other formats" id="oth-2411.16589" aria-labelledby="oth-2411.16589">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> The Grassmann distance complexity </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Lerario,+A">Antonio Lerario</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Rosana,+A">Andrea Rosana</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 41 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Differential Geometry (math.DG)</span>; Algebraic Geometry (math.AG) </div> <p class='mathjax'> Motivated by the concept of Euclidean Distance Degree, which measures the complexity of finding the nearest point to an algebraic set in Euclidean space, we introduce the notion of Grassmann Distance Complexity (GDC). This concept quantifies the complexity of solving the nearest point problem for subanalytic sets in the Grassmannian, using the intrinsic Riemannian distance. Unlike the Euclidean case, the Grassmannian distance is neither smooth nor semialgebraic, and its study requires using Lipschitz critical point theory and o-minimal geometry. We establish fundamental properties of GDC, including computable bounds for real algebraic varieties and conditions ensuring the finiteness of critical points. Our results also include a nonlinear version of the classical Eckart-Young theorem, which characterizes critical points of the distance function from a generic $k$-plane to simple Schubert varieties. </p> </div> </dd> </dl> <dl id='articles'> <h3>Replacement submissions (showing 22 of 22 entries)</h3> <dt> <a name='item22'>[22]</a> <a href ="/abs/2105.15090" title="Abstract" id="2105.15090"> arXiv:2105.15090 </a> (replaced) [<a href="/pdf/2105.15090" title="Download PDF" id="pdf-2105.15090" aria-labelledby="pdf-2105.15090">pdf</a>, <a href="https://arxiv.org/html/2105.15090v2" title="View HTML" id="html-2105.15090" aria-labelledby="html-2105.15090" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2105.15090" title="Other formats" id="oth-2105.15090" aria-labelledby="oth-2105.15090">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Uniformity of quadratic points </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Ge,+T">Tangli Ge</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 25 pages; revised final version </div> <div class='list-journal-ref'><span class='descriptor'>Journal-ref:</span> International Journal of Number Theory, Vol. 20, No. 04, pp. 1041-1071 (2024) </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Number Theory (math.NT) </div> <p class='mathjax'> In this paper, we extend a uniformity result of Dimitrov-Gao-Habegger to dimension two and use it to get a uniform bound on the set of all quadratic points for non-hyperelliptic non-bielliptic curves in terms of the Mordell-Weil rank. </p> </div> </dd> <dt> <a name='item23'>[23]</a> <a href ="/abs/2108.12785" title="Abstract" id="2108.12785"> arXiv:2108.12785 </a> (replaced) [<a href="/pdf/2108.12785" title="Download PDF" id="pdf-2108.12785" aria-labelledby="pdf-2108.12785">pdf</a>, <a href="/format/2108.12785" title="Other formats" id="oth-2108.12785" aria-labelledby="oth-2108.12785">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On the cohomology of p-adic analytic spaces, II: The $C_{\rm st}$-conjecture </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Colmez,+P">Pierre Colmez</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Nizio%C5%82,+W">Wies艂awa Nizio艂</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> Final version. To appear in Duke Math. J </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Number Theory (math.NT) </div> <p class='mathjax'> Long ago, Fontaine formulated conjectures (now theorems) relating 茅tale and de Rham cohomologies of algebraic varieties over $p$-adic fields. In an earlier work we have shown that pro-茅tale and de Rham cohomologies of analytic varieties in the two extreme cases: proper and Stein, are also related. In the proper case, the comparison theorems are similar to those for algebraic varieties, but for Stein varieties they are quite different. <br>In this paper, we state analogs of Fontaine's conjectures for general smooth dagger varieties, that interpolate between these two extreme cases, and we prove them for many ``small'' varieties (cases we deal with include products of overconvergent affinoids and proper varieties, analytifications of algebraic varieties, or "almost proper" dagger varieties). The proofs use a ``geometrization'' of all involved cohomologies in terms of quasi-Banach-Colmez spaces (qBC's for short, quasi- because we relax the finiteness conditions). The heart of the proof relies on delicate properties of BC's and qBC's. These properties should be of independent interest and we have devoted a large part of the paper to them. </p> </div> </dd> <dt> <a name='item24'>[24]</a> <a href ="/abs/2205.09926" title="Abstract" id="2205.09926"> arXiv:2205.09926 </a> (replaced) [<a href="/pdf/2205.09926" title="Download PDF" id="pdf-2205.09926" aria-labelledby="pdf-2205.09926">pdf</a>, <a href="https://arxiv.org/html/2205.09926v4" title="View HTML" id="html-2205.09926" aria-labelledby="html-2205.09926" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2205.09926" title="Other formats" id="oth-2205.09926" aria-labelledby="oth-2205.09926">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Smoothing, scattering, and a conjecture of Fukaya </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Chan,+K">Kwokwai Chan</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Leung,+N+C">Naichung Conan Leung</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Ma,+Z+N">Ziming Nikolas Ma</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 65 pages, 9 figures; v4: final version </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Symplectic Geometry (math.SG) </div> <p class='mathjax'> In 2002, Fukaya proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi-Yau manifold $\check{X}$ and the multi-valued Morse theory on the base $\check{B}$ of an SYZ fibration $\check{p}: \check{X}\to \check{B}$, and the other between deformation theory of the mirror $X$ and the same multi-valued Morse theory on $\check{B}$. In this paper, we prove a reformulation of the main conjecture in Fukaya's second correspondence, where multi-valued Morse theory on the base $\check{B}$ is replaced by tropical geometry on the Legendre dual $B$. In the proof, we apply techniques of asymptotic analysis developed in our previous works to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety introduced in another of our recent work. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part $X_{\text{sf}} \subseteq X$ allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions. </p> </div> </dd> <dt> <a name='item25'>[25]</a> <a href ="/abs/2208.03254" title="Abstract" id="2208.03254"> arXiv:2208.03254 </a> (replaced) [<a href="/pdf/2208.03254" title="Download PDF" id="pdf-2208.03254" aria-labelledby="pdf-2208.03254">pdf</a>, <a href="https://arxiv.org/html/2208.03254v3" title="View HTML" id="html-2208.03254" aria-labelledby="html-2208.03254" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2208.03254" title="Other formats" id="oth-2208.03254" aria-labelledby="oth-2208.03254">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> A Serre-type spectral sequence for motivic cohomology </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Tanania,+F">Fabio Tanania</a></div> <div class='list-journal-ref'><span class='descriptor'>Journal-ref:</span> Algebr. Geom. 11 (2024), no. 3, 386-420 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Algebraic Topology (math.AT); K-Theory and Homology (math.KT) </div> <p class='mathjax'> In this paper, we construct and study a Serre-type spectral sequence for motivic cohomology associated to a map of bisimplicial schemes with motivically cellular fiber. Then, we show how to apply it in order to approach the computation of the motivic cohomology of the Nisnevich classifying space of projective general linear groups. This naturally yields an explicit description of the motive of a Severi-Brauer variety in terms of twisted motives of its 膶ech simplicial scheme. </p> </div> </dd> <dt> <a name='item26'>[26]</a> <a href ="/abs/2302.04772" title="Abstract" id="2302.04772"> arXiv:2302.04772 </a> (replaced) [<a href="/pdf/2302.04772" title="Download PDF" id="pdf-2302.04772" aria-labelledby="pdf-2302.04772">pdf</a>, <a href="https://arxiv.org/html/2302.04772v3" title="View HTML" id="html-2302.04772" aria-labelledby="html-2302.04772" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2302.04772" title="Other formats" id="oth-2302.04772" aria-labelledby="oth-2302.04772">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Motivic cohomology of the Nisnevich classifying space of even Clifford groups </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Tanania,+F">Fabio Tanania</a></div> <div class='list-journal-ref'><span class='descriptor'>Journal-ref:</span> Doc. Math. 29 (2024), no. 1, 191-208 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Algebraic Topology (math.AT); K-Theory and Homology (math.KT) </div> <p class='mathjax'> In this paper, we consider the split even Clifford group $\Gamma^+_n$ and compute the mod 2 motivic cohomology ring of its Nisnevich classifying space. The description we obtain is quite similar to the one provided for spin groups in [11]. The fundamental difference resides in the behaviour of the second subtle Stiefel-Whitney class that is non-trivial for even Clifford groups, while it vanished in the spin-case. </p> </div> </dd> <dt> <a name='item27'>[27]</a> <a href ="/abs/2304.12755" title="Abstract" id="2304.12755"> arXiv:2304.12755 </a> (replaced) [<a href="/pdf/2304.12755" title="Download PDF" id="pdf-2304.12755" aria-labelledby="pdf-2304.12755">pdf</a>, <a href="https://arxiv.org/html/2304.12755v3" title="View HTML" id="html-2304.12755" aria-labelledby="html-2304.12755" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2304.12755" title="Other formats" id="oth-2304.12755" aria-labelledby="oth-2304.12755">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Cylindrical ample divisors on Du Val del Pezzo surfaces </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Sawahara,+M">Masatomo Sawahara</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 23 pages, 8 figures; v3: Many typos are revised. To appear in Forum Mathematicum </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> Let $S$ be a del Pezzo surface with at worse Du Val singularities of degree $\ge 3$. We construct an $H$-polar cylinder for any ample $\mathbb{Q}$-divisor $H$ on $S$. </p> </div> </dd> <dt> <a name='item28'>[28]</a> <a href ="/abs/2310.06058" title="Abstract" id="2310.06058"> arXiv:2310.06058 </a> (replaced) [<a href="/pdf/2310.06058" title="Download PDF" id="pdf-2310.06058" aria-labelledby="pdf-2310.06058">pdf</a>, <a href="/format/2310.06058" title="Other formats" id="oth-2310.06058" aria-labelledby="oth-2310.06058">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Gromov-Witten theory of bicyclic pairs </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=van+Garrel,+M">Michel van Garrel</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Nabijou,+N">Navid Nabijou</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Schuler,+Y">Yannik Schuler</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 42 pages. Added an application of the main result </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> A bicyclic pair is a smooth surface equipped with a pair of smooth divisors intersecting in two reduced points. Resolutions of self-nodal curves constitute an important special case. We investigate the logarithmic Gromov-Witten theory of bicyclic pairs. We establish correspondences with local Gromov-Witten theory and open Gromov-Witten theory in all genera, a correspondence with orbifold Gromov-Witten theory in genus zero, and correspondences between all-genus refined Gopakumar-Vafa invariants and refined quiver Donaldson-Thomas invariants. For self-nodal curves in $\mathbb{P}(1,1,r)$ we obtain closed formulae for the genus zero invariants and relate these to the invariants of local curves. We also establish a conceptual relationship between invariants relative a self-nodal plane cubic and invariants relative a smooth plane cubic. The technical heart of the paper is a qualitatively new analysis of the degeneration formula for stable logarithmic maps, involving a tight intertwining of tropical and intersection-theoretic vanishing arguments. </p> </div> </dd> <dt> <a name='item29'>[29]</a> <a href ="/abs/2310.07666" title="Abstract" id="2310.07666"> arXiv:2310.07666 </a> (replaced) [<a href="/pdf/2310.07666" title="Download PDF" id="pdf-2310.07666" aria-labelledby="pdf-2310.07666">pdf</a>, <a href="https://arxiv.org/html/2310.07666v2" title="View HTML" id="html-2310.07666" aria-labelledby="html-2310.07666" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2310.07666" title="Other formats" id="oth-2310.07666" aria-labelledby="oth-2310.07666">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Torelli theorem for moduli stacks of vector bundles and principal G-bundles </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Alfaya,+D">David Alfaya</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Biswas,+I">Indranil Biswas</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=G%C3%B3mez,+T+L">Tom谩s L. G贸mez</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Mukhopadhyay,+S">Swarnava Mukhopadhyay</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 19 pages </div> <div class='list-journal-ref'><span class='descriptor'>Journal-ref:</span> Journal of Geometry and Physics 207 (2025) 105350 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> Given any irreducible smooth complex projective curve $X$, of genus at least $2$, consider the moduli stack of vector bundles on $X$ of fixed rank and determinant. It is proved that the isomorphism class of the stack uniquely determines the isomorphism class of the curve $X$ and the rank of the vector bundles. The case of trivial determinant, rank $2$ and genus $2$ is specially interesting: the curve can be recovered from the moduli stack, but not from the moduli space (since this moduli space is $\mathbb{P}^3$ thus independently of the curve). <br>We also prove a Torelli theorem for moduli stacks of principal $G$-bundles on a curve of genus at least $3$, where $G$ is any non-abelian reductive group. </p> </div> </dd> <dt> <a name='item30'>[30]</a> <a href ="/abs/2401.04298" title="Abstract" id="2401.04298"> arXiv:2401.04298 </a> (replaced) [<a href="/pdf/2401.04298" title="Download PDF" id="pdf-2401.04298" aria-labelledby="pdf-2401.04298">pdf</a>, <a href="https://arxiv.org/html/2401.04298v2" title="View HTML" id="html-2401.04298" aria-labelledby="html-2401.04298" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2401.04298" title="Other formats" id="oth-2401.04298" aria-labelledby="oth-2401.04298">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Fibered Calabi-Yau threefolds with relative automorphisms of positive entropy and $c_2$-contractions </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Oguiso,+K">Keiji Oguiso</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 17 Pages, a fairly major revision after the referee's suggestions and questions, final form accepted by RCMP (Rendiconti del Circolo Matematico di Palermo Series 2) </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> We show that an abelian fibered Calabi-Yau threefold with a positive entropy automorphism preserving the fibration is unique up to isomorphisms as fibered varieties. We also give a fairly explicit structure theorem of an elliptically fibered Calabi-Yau threefold with a positive entropy automorphism preserving the fibration. </p> </div> </dd> <dt> <a name='item31'>[31]</a> <a href ="/abs/2402.18955" title="Abstract" id="2402.18955"> arXiv:2402.18955 </a> (replaced) [<a href="/pdf/2402.18955" title="Download PDF" id="pdf-2402.18955" aria-labelledby="pdf-2402.18955">pdf</a>, <a href="/format/2402.18955" title="Other formats" id="oth-2402.18955" aria-labelledby="oth-2402.18955">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Santal\'o Geometry of Convex Polytopes </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Pavlov,+D">Dmitrii Pavlov</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Telen,+S">Simon Telen</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 27 pages, 4 figures, comments welcome </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Optimization and Control (math.OC) </div> <p class='mathjax'> The Santal贸 point of a convex polytope is the interior point which leads to a polar dual of minimal volume. This minimization problem is relevant in interior point methods for convex optimization, where the logarithm of the dual volume is known as the universal barrier function. When translating the facet hyperplanes, the Santal贸 point traces out a semi-algebraic set. We describe and compute this geometry using algebraic and numerical techniques. We exploit connections with statistics, optimization and physics. </p> </div> </dd> <dt> <a name='item32'>[32]</a> <a href ="/abs/2403.13377" title="Abstract" id="2403.13377"> arXiv:2403.13377 </a> (replaced) [<a href="/pdf/2403.13377" title="Download PDF" id="pdf-2403.13377" aria-labelledby="pdf-2403.13377">pdf</a>, <a href="https://arxiv.org/html/2403.13377v2" title="View HTML" id="html-2403.13377" aria-labelledby="html-2403.13377" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2403.13377" title="Other formats" id="oth-2403.13377" aria-labelledby="oth-2403.13377">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Singular plane curves: freeness and combinatorics </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Cuntz,+M">Michael Cuntz</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Pokora,+P">Piotr Pokora</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 17 pages, 2 figures, Michael Cuntz joined as a co-author, new title </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Combinatorics (math.CO) </div> <p class='mathjax'> In this paper we focus on various aspects of singular complex plane curves, mostly in the context of their homological properties and the associated combinatorial structures. We formulate some challenging open problems that can point to new directions in research, for example by introducing weak Ziegler pairs of curve arrangements. Moreover, we construct new examples of different Ziegler pairs, in both the classical and the weak sense, and present new geometric approaches to construction problems of singular plane curves. </p> </div> </dd> <dt> <a name='item33'>[33]</a> <a href ="/abs/2405.13472" title="Abstract" id="2405.13472"> arXiv:2405.13472 </a> (replaced) [<a href="/pdf/2405.13472" title="Download PDF" id="pdf-2405.13472" aria-labelledby="pdf-2405.13472">pdf</a>, <a href="https://arxiv.org/html/2405.13472v2" title="View HTML" id="html-2405.13472" aria-labelledby="html-2405.13472" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2405.13472" title="Other formats" id="oth-2405.13472" aria-labelledby="oth-2405.13472">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On the geometry of singular EPW cubes </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Rizzo,+F">Francesca Rizzo</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> Version two, comments are welcome! </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> EPW cubes form a locally complete family of smooth projective hyper-K盲hler varieties of dimension 6, constructed by Iliev--Kapustka--Kapustka--Ranestad.\ Their construction and behavior share a lot of similarities with the double EPW sextics constructed by O'Grady.\ Adapting the methods of O'Grady, we construct a projective smooth small resolution of singular EPW cubes. </p> </div> </dd> <dt> <a name='item34'>[34]</a> <a href ="/abs/2406.02922" title="Abstract" id="2406.02922"> arXiv:2406.02922 </a> (replaced) [<a href="/pdf/2406.02922" title="Download PDF" id="pdf-2406.02922" aria-labelledby="pdf-2406.02922">pdf</a>, <a href="/format/2406.02922" title="Other formats" id="oth-2406.02922" aria-labelledby="oth-2406.02922">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Saturated de Rham-Witt complexes with unit-root coefficients </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Fernando,+R">Ravi Fernando</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 75 pages; condensed the preliminary material and made various local changes </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> The saturated de Rham-Witt complex, introduced by Bhatt-Lurie-Mathew, is a variant of the classical de Rham-Witt complex which provides a conceptual simplification of the construction and which is expected to produce better results for non-smooth varieties. In this paper, we introduce a generalization of the saturated de Rham-Witt complex which allows coefficients in a unit-root $F$-crystal. We define our complex by a universal property in a category of so-called de Rham-Witt modules. We prove a number of results about it, including existence, quasicoherence, and comparisons to the de Rham-Witt complex of Bhatt-Lurie-Mathew and (in the smooth case) to crystalline cohomology and the classical de Rham-Witt complex with coefficients. </p> </div> </dd> <dt> <a name='item35'>[35]</a> <a href ="/abs/2407.08492" title="Abstract" id="2407.08492"> arXiv:2407.08492 </a> (replaced) [<a href="/pdf/2407.08492" title="Download PDF" id="pdf-2407.08492" aria-labelledby="pdf-2407.08492">pdf</a>, <a href="https://arxiv.org/html/2407.08492v3" title="View HTML" id="html-2407.08492" aria-labelledby="html-2407.08492" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2407.08492" title="Other formats" id="oth-2407.08492" aria-labelledby="oth-2407.08492">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Syzygies of general projections of canonical and paracanonical curves </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Li,+L">Li Li</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 22 pages. Add the explanations on why the problem cannot be solved by deformation to the inner projection. Add the criterion on the reducedness </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> Let $X\subset\mathbb{P}^r$ be an integral linearly normal variety and $R=k[x_0,\cdots,x_r]$ the coordinate ring of $\mathbb{P}^r$. It is known that the syzygies of $X$ contain some geometric information. In recent years the syzygies of non-projectively normal varieties or in other words, the projection $X'$ of $X$ away from a linear subspace $W\subset\mathbb{P}^r$, were taken into considerations. Assuming that the coordinate ring of the ambient space that $X'$ lives in is $S$, there are two types of vanishing properties of the Betti diagrams of the projected varieties, the so-called $N_{d,p}^S$ and $\widetilde{N}_{d,p}$. The former one have been widely discussed for general varieties, for example by S. Kwak, Y. Choi and E. Park, while the latter one was discussed by W. Lee and E. Park for curves of very large degree. <br>In this paper I will discuss about the $\widetilde{N}_{d,p}$ properties of the projection of a generic canonical and paracanonical curve away from a generic point and in particular whether they are cut out by quadrics. Some conjectures will be claimed based on the tests on Macaulay2. </p> </div> </dd> <dt> <a name='item36'>[36]</a> <a href ="/abs/2407.09178" title="Abstract" id="2407.09178"> arXiv:2407.09178 </a> (replaced) [<a href="/pdf/2407.09178" title="Download PDF" id="pdf-2407.09178" aria-labelledby="pdf-2407.09178">pdf</a>, <a href="https://arxiv.org/html/2407.09178v3" title="View HTML" id="html-2407.09178" aria-labelledby="html-2407.09178" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2407.09178" title="Other formats" id="oth-2407.09178" aria-labelledby="oth-2407.09178">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Shafarevich-Tate groups of holomorphic Lagrangian fibrations II </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Abasheva,+A">Anna Abasheva</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> Fixed a mistake in the first version. Removed the assumptions of the smoothness of the base of a Lagrangian fibrations from all statements </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Complex Variables (math.CV) </div> <p class='mathjax'> Let $X$ be a compact hyperk盲hler manifold with a Lagrangian fibration $\pi\colon X\to B$. A Shafarevich-Tate twist of $X$ is a holomorphic symplectic manifold with a Lagrangian fibration $\pi^\varphi\colon X^\varphi\to B$ which is isomorphic to $\pi$ locally over the base. In particular, $\pi^\varphi$ has the same fibers as $\pi$. A twist $X^\varphi$ corresponds to an element $\varphi$ in the Shafarevich-Tate group of $X$. We show that $X^\varphi$ is K盲hler when a multiple of $\varphi$ lies in the connected component of unity of the Shafarevich-Tate group and give a necessary condition for $X^\varphi$ to be bimeromorphic to a K盲hler manifold. </p> </div> </dd> <dt> <a name='item37'>[37]</a> <a href ="/abs/2407.19870" title="Abstract" id="2407.19870"> arXiv:2407.19870 </a> (replaced) [<a href="/pdf/2407.19870" title="Download PDF" id="pdf-2407.19870" aria-labelledby="pdf-2407.19870">pdf</a>, <a href="https://arxiv.org/html/2407.19870v2" title="View HTML" id="html-2407.19870" aria-labelledby="html-2407.19870" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2407.19870" title="Other formats" id="oth-2407.19870" aria-labelledby="oth-2407.19870">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Optimal upper bounds for anti-canonical volumes of singular toric Fano varieties </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Zou,+Y">Yu Zou</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 31 pages, minor correction, some improvement on the main result; comments are welcome </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> Fix two positive integers $d\geq3$ and $q$. We give an upper bound for anti-canonical volumes of $d$-dimensional $\frac{1}{q}$-lc toric Fano varieties, which corresponds to an upper bound for the dual normalized volumes of the associated $d$-dimensional $\frac{1}{q}$-lc Fano polytopes. And we also construct examples to show that these upper bounds are optimal. Besides, we provide an optimal upper bound for volumes of $d$-dimensional lattice simplices $S$ such that $\frac{1}{q}S$ has exactly one interior lattice point. </p> </div> </dd> <dt> <a name='item38'>[38]</a> <a href ="/abs/2411.07129" title="Abstract" id="2411.07129"> arXiv:2411.07129 </a> (replaced) [<a href="/pdf/2411.07129" title="Download PDF" id="pdf-2411.07129" aria-labelledby="pdf-2411.07129">pdf</a>, <a href="/format/2411.07129" title="Other formats" id="oth-2411.07129" aria-labelledby="oth-2411.07129">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> A solution to Fujita's freeness conjecture via an extension theorem with analytic adjoint ideal sheaves </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Chan,+T+O+M">Tsz On Mario Chan</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> Thanks to Prof. S. Helmke for pointing out that the construction of $f^{\text{loc}}$ on p.31 need not yield a local hol. section with the desired high vanishing order. Example: when $\mathtt{Z}$ locally behaves like $\{xy = z^2\}$, on which $x =\frac{z^2}{y}$ has vanishing order $2$ along $\{x=z=0\}$. It cannot be hol. extended to the ambient space locally with the vanishing order preserved </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Complex Variables (math.CV) </div> <p class='mathjax'> The effective freeness in Fujita's conjecture states that, for an ample line bundle $L$ on a complex projective manifold $X$, the adjoint bundle $K_X\otimes L^{\otimes m}$ is globally generated when $m \geq \dim_{\mathbb C} X + 1$. Following the approach of Angehrn and Siu, a solution is provided in this paper via the use of adjoint ideal sheaves, which provide a finer control of the non-integrable loci given by multiplier ideal sheaves, so that one can work directly with the lc singularities and the associated (minimal) lc centres as in the algebraic approaches of Kawamata and Helmke. The substitute for the Nadel or Kawamata-Viehweg vanishing theorem used in previous approaches is an extension theorem based on the techniques developed for the injectivity theorems. </p> </div> </dd> <dt> <a name='item39'>[39]</a> <a href ="/abs/2411.10763" title="Abstract" id="2411.10763"> arXiv:2411.10763 </a> (replaced) [<a href="/pdf/2411.10763" title="Download PDF" id="pdf-2411.10763" aria-labelledby="pdf-2411.10763">pdf</a>, <a href="https://arxiv.org/html/2411.10763v2" title="View HTML" id="html-2411.10763" aria-labelledby="html-2411.10763" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.10763" title="Other formats" id="oth-2411.10763" aria-labelledby="oth-2411.10763">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Canonical blow-ups of Grassmannians I: How canonical is a Kausz compactification? </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Fang,+H">Hanlong Fang</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Wu,+X">Xian Wu</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> A funding source is added, and some typos are corrected. All comments are welcome! </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span> </div> <p class='mathjax'> In this paper, we develop a simple uniform picture incorporating the Kausz compactifications and the spaces of complete collineations by blowing up Grassmannians $G(p,n)$ according to a torus action $\mathbb G_m$. We show that each space of complete collineations is isomorphic to any maximal-dimensional connected component of the $\mathbb G_m$-fixed point scheme of a Kausz-type compactification. We prove that the Kausz-type compactification is the total family over the Hilbert quotient $G(p,n)/ \! \! / \mathbb G_m$ which is isomorphic to the space of complete collineations. In particular, the Kausz compactifications are toroidal embeddings of general linear groups in the sense of Brion-Kumar. We also show that the Kausz-type compactifications resolve the Landsberg-Manivel birational maps from projective spaces to Grassmannians, by comparing Kausz's construction with ours. As an application, by studying the foliation we derive resolutions of certain birational maps among projective bundles over Grassmannians. The results in this paper are partially taken from the first author's earlier arxiv post (Canonical blow-ups of grassmann manifolds, <a href="https://arxiv.org/abs/2007.06200" data-arxiv-id="2007.06200" class="link-https">arXiv:2007.06200</a>), which has been revised and expanded in collaboration with the second author. </p> </div> </dd> <dt> <a name='item40'>[40]</a> <a href ="/abs/2411.12500" title="Abstract" id="2411.12500"> arXiv:2411.12500 </a> (replaced) [<a href="/pdf/2411.12500" title="Download PDF" id="pdf-2411.12500" aria-labelledby="pdf-2411.12500">pdf</a>, <a href="https://arxiv.org/html/2411.12500v2" title="View HTML" id="html-2411.12500" aria-labelledby="html-2411.12500" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.12500" title="Other formats" id="oth-2411.12500" aria-labelledby="oth-2411.12500">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Weyl group of type $E_6$ and K3 surfaces </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Bonnaf%C3%A9,+C">C茅dric Bonnaf茅</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 46 pages, Magma codes included </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Representation Theory (math.RT) </div> <p class='mathjax'> Adapting methods of previous papers by A. Sarti and the author, we construct K3 surfaces from invariants of the Weyl group of type $E_6$. We study in details one of these surfaces, which turns out to have Picard number $20$: for this example, we describe an elliptic fibration (and its singular fibers) and the Picard lattice. </p> </div> </dd> <dt> <a name='item41'>[41]</a> <a href ="/abs/2103.07422" title="Abstract" id="2103.07422"> arXiv:2103.07422 </a> (replaced) [<a href="/pdf/2103.07422" title="Download PDF" id="pdf-2103.07422" aria-labelledby="pdf-2103.07422">pdf</a>, <a href="https://arxiv.org/html/2103.07422v4" title="View HTML" id="html-2103.07422" aria-labelledby="html-2103.07422" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2103.07422" title="Other formats" id="oth-2103.07422" aria-labelledby="oth-2103.07422">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Distinguished categories and the Zilber-Pink conjecture </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Barroero,+F">Fabrizio Barroero</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Dill,+G+A">Gabriel Andreas Dill</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 50 pages, to appear in Amer. J. Math </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Number Theory (math.NT)</span>; Algebraic Geometry (math.AG) </div> <p class='mathjax'> We propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mixed Shimura varieties. It allows us to define all basic concepts of the field and prove some fundamental facts about them, e.g. the defect condition. <br>In some categories that we call very distinguished, we are able to show some implications between Zilber-Pink statements with respect to base change. This yields unconditional results, i.e. the Zilber-Pink conjecture for a complex curve in $\mathcal{A}_2$ that cannot be defined over $\bar{\mathbb{Q}}$, a complex curve in the $g$-th fibered power of the Legendre family, and a complex curve in the base change of a semiabelian variety over $\bar{\mathbb{Q}}$. </p> </div> </dd> <dt> <a name='item42'>[42]</a> <a href ="/abs/2202.02497" title="Abstract" id="2202.02497"> arXiv:2202.02497 </a> (replaced) [<a href="/pdf/2202.02497" title="Download PDF" id="pdf-2202.02497" aria-labelledby="pdf-2202.02497">pdf</a>, <a href="https://arxiv.org/html/2202.02497v2" title="View HTML" id="html-2202.02497" aria-labelledby="html-2202.02497" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2202.02497" title="Other formats" id="oth-2202.02497" aria-labelledby="oth-2202.02497">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Uniform Mordell-Lang Plus Bogomolov </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Ge,+T">Tangli Ge</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 14 pages; revised final version </div> <div class='list-journal-ref'><span class='descriptor'>Journal-ref:</span> International Mathematics Research Notices, Volume 2024, Issue 9, May 2024 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Number Theory (math.NT)</span>; Algebraic Geometry (math.AG) </div> <p class='mathjax'> In this paper, we prove a uniform version of Poonen's "Mordell-Lang Plus Bogomolov" theorem for abelian varieties. We mainly generalize R茅mond's work on large points to allow an extra $\epsilon$-neighborhood. The part on small points follows from an earlier paper, joint with Gao and K眉hne. </p> </div> </dd> <dt> <a name='item43'>[43]</a> <a href ="/abs/2403.13637" title="Abstract" id="2403.13637"> arXiv:2403.13637 </a> (replaced) [<a href="/pdf/2403.13637" title="Download PDF" id="pdf-2403.13637" aria-labelledby="pdf-2403.13637">pdf</a>, <a href="https://arxiv.org/html/2403.13637v4" title="View HTML" id="html-2403.13637" aria-labelledby="html-2403.13637" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2403.13637" title="Other formats" id="oth-2403.13637" aria-labelledby="oth-2403.13637">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> DG singular equivalence and singular locus </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Liu,+L">Leilei Liu</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Zeng,+J">Jieheng Zeng</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 18 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Commutative Algebra (math.AC)</span>; Algebraic Geometry (math.AG); Rings and Algebras (math.RA); Representation Theory (math.RT) </div> <p class='mathjax'> For a commutative Gorenstein Noetherian ring $R$, we construct an affine scheme $X$ solely from DG singularity category $S_{dg}(R)$ of $R$ such that there is a finite surjective morphism $X \rightarrow \mathrm{Spec}(R /I)$, where $\mathrm{Spec}(R /I)$ is the singular locus in $\mathrm{Spec}(R)$. As an application, for two such rings with equivalent DG singularity categories, we prove that the singular loci in their affine schemes have the same dimension. </p> </div> </dd> </dl> <div class='paging'>Total of 43 entries </div> <div class='morefewer'>Showing up to 2000 entries per page: <a href=/list/math.AG/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> </div> </div> </div> </main> <footer style="clear: both;"> <div class="columns is-desktop" role="navigation" aria-label="Secondary" style="margin: -0.75em -0.75em 0.75em -0.75em">  <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div>   <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; 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