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class="control is-expanded"> <label for="query" class="hidden-label">Search term or terms</label> <input class="input is-medium" id="query" name="query" placeholder="Search term..." type="text" value="Krug, A"> </div> <div class="select control is-medium"> <label class="is-hidden" for="searchtype">Field</label> <select class="is-medium" id="searchtype" name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author 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name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Krug, A"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" type="radio" value="hide"> <label for="abstracts-1">Hide abstracts</label></li></ul> </div> <div class="box field is-grouped is-grouped-multiline level-item"> <div class="control"> <span class="select is-small"> <select id="size" name="size"><option value="25">25</option><option selected value="50">50</option><option value="100">100</option><option value="200">200</option></select> </span> <label for="size">results per page</label>. </div> <div class="control"> <label for="order">Sort results by</label> <span class="select is-small"> <select id="order" name="order"><option selected value="-announced_date_first">Announcement date (newest first)</option><option value="announced_date_first">Announcement date (oldest first)</option><option value="-submitted_date">Submission date (newest first)</option><option value="submitted_date">Submission date (oldest first)</option><option value="">Relevance</option></select> </span> </div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.08991">arXiv:2409.08991</a> <span> [<a href="https://arxiv.org/pdf/2409.08991">pdf</a>, <a href="https://arxiv.org/ps/2409.08991">ps</a>, <a href="https://arxiv.org/format/2409.08991">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> A smooth but non-symplectic moduli of sheaves on a hyperk盲hler variety </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a>, <a href="/search/math?searchtype=author&query=Reede%2C+F">Fabian Reede</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Z">Ziyu Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.08991v1-abstract-short" style="display: inline;"> For an abelian surface $A$, we consider stable vector bundles on a generalized Kummer variety $K_n(A)$ with $n>1$. We prove that the connected component of the moduli space which contains the tautological bundles associated to line bundles of degree $0$ is isomorphic to the blowup of the dual abelian surface in one point. We believe that this is the first explicit example of a component which is s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.08991v1-abstract-full').style.display = 'inline'; document.getElementById('2409.08991v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.08991v1-abstract-full" style="display: none;"> For an abelian surface $A$, we consider stable vector bundles on a generalized Kummer variety $K_n(A)$ with $n>1$. We prove that the connected component of the moduli space which contains the tautological bundles associated to line bundles of degree $0$ is isomorphic to the blowup of the dual abelian surface in one point. We believe that this is the first explicit example of a component which is smooth with a non-trivial canonical bundle. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.08991v1-abstract-full').style.display = 'none'; document.getElementById('2409.08991v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages. Comments are welcome!</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.19814">arXiv:2403.19814</a> <span> [<a href="https://arxiv.org/pdf/2403.19814">pdf</a>, <a href="https://arxiv.org/ps/2403.19814">ps</a>, <a href="https://arxiv.org/format/2403.19814">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> </div> </div> <p class="title is-5 mathjax"> Endomorphism Algebras of Equivariant Exceptional Collections </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a>, <a href="/search/math?searchtype=author&query=Nikolov%2C+E">Erik Nikolov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.19814v1-abstract-short" style="display: inline;"> Given an action of a finite group on a triangulated category with a suitable strong exceptional collection, a construction of Elagin produces an associated strong exceptional collection on the equivariant category. We prove that the endomorphism algebra of the induced exceptional collection is the basic reduction of the skew group algebra of the endomorphism algebra of the original exceptional col… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.19814v1-abstract-full').style.display = 'inline'; document.getElementById('2403.19814v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.19814v1-abstract-full" style="display: none;"> Given an action of a finite group on a triangulated category with a suitable strong exceptional collection, a construction of Elagin produces an associated strong exceptional collection on the equivariant category. We prove that the endomorphism algebra of the induced exceptional collection is the basic reduction of the skew group algebra of the endomorphism algebra of the original exceptional collection. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.19814v1-abstract-full').style.display = 'none'; document.getElementById('2403.19814v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.06244">arXiv:2309.06244</a> <span> [<a href="https://arxiv.org/pdf/2309.06244">pdf</a>, <a href="https://arxiv.org/format/2309.06244">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Hochschild cohomology of Hilbert schemes of points on surfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Belmans%2C+P">Pieter Belmans</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2309.06244v2-abstract-short" style="display: inline;"> We compute the Hochschild cohomology of Hilbert schemes of points on surfaces and observe that it is, in general, not determined solely by the Hochschild cohomology of the surface, but by its "Hochschild-Serre cohomology": the bigraded vector space obtained by taking Hochschild homologies with coefficients in powers of the Serre functor. As applications, we obtain various consequences on the defor… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.06244v2-abstract-full').style.display = 'inline'; document.getElementById('2309.06244v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.06244v2-abstract-full" style="display: none;"> We compute the Hochschild cohomology of Hilbert schemes of points on surfaces and observe that it is, in general, not determined solely by the Hochschild cohomology of the surface, but by its "Hochschild-Serre cohomology": the bigraded vector space obtained by taking Hochschild homologies with coefficients in powers of the Serre functor. As applications, we obtain various consequences on the deformation theory of the Hilbert schemes; in particular, we recover and extend results of Fantechi, Boissi猫re, and Hitchin. Our method is to compute more generally for any smooth proper algebraic variety $X$ the Hochschild-Serre cohomology of the symmetric quotient stack $[X^n/\mathfrak{S}_n]$, in terms of the Hochschild-Serre cohomology of $X$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.06244v2-abstract-full').style.display = 'none'; document.getElementById('2309.06244v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">45 pages, added further evidence for conjecture</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.17124">arXiv:2305.17124</a> <span> [<a href="https://arxiv.org/pdf/2305.17124">pdf</a>, <a href="https://arxiv.org/ps/2305.17124">ps</a>, <a href="https://arxiv.org/format/2305.17124">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Extension Groups of Tautological Bundles on Punctual Quot Schemes of Curves </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2305.17124v2-abstract-short" style="display: inline;"> We prove formulas for the cohomology and the extension groups of tautological bundles on punctual Quot schemes over complex smooth projective curves. As a corollary, we show that the tautological bundle determines the isomorphism class of the original vector bundle on the curve. We also give a vanishing result for the push-forward along the Quot--Chow morphism of tensor and wedge products of duals… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.17124v2-abstract-full').style.display = 'inline'; document.getElementById('2305.17124v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.17124v2-abstract-full" style="display: none;"> We prove formulas for the cohomology and the extension groups of tautological bundles on punctual Quot schemes over complex smooth projective curves. As a corollary, we show that the tautological bundle determines the isomorphism class of the original vector bundle on the curve. We also give a vanishing result for the push-forward along the Quot--Chow morphism of tensor and wedge products of duals of tautological bundles. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.17124v2-abstract-full').style.display = 'none'; document.getElementById('2305.17124v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Small changes in exposition, fixed some typos</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.03436">arXiv:2303.03436</a> <span> [<a href="https://arxiv.org/pdf/2303.03436">pdf</a>, <a href="https://arxiv.org/ps/2303.03436">ps</a>, <a href="https://arxiv.org/format/2303.03436">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> </div> </div> <p class="title is-5 mathjax"> Asymmetry of $\mathbb P$-Functors </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hochenegger%2C+A">Andreas Hochenegger</a>, <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2303.03436v1-abstract-short" style="display: inline;"> Recently, a new definition of $\mathbb P$-functors was proposed by Anno and Logvinenko. In their article, the authors wonder whether this notion is symmetric in the sense that the adjoints of $\mathbb P$-functors are again $\mathbb P$-functors, the analogue being true for spherical functors. We give geometric examples involving the Hilbert scheme of points on a surface that yield a negative answer… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.03436v1-abstract-full').style.display = 'inline'; document.getElementById('2303.03436v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.03436v1-abstract-full" style="display: none;"> Recently, a new definition of $\mathbb P$-functors was proposed by Anno and Logvinenko. In their article, the authors wonder whether this notion is symmetric in the sense that the adjoints of $\mathbb P$-functors are again $\mathbb P$-functors, the analogue being true for spherical functors. We give geometric examples involving the Hilbert scheme of points on a surface that yield a negative answer. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.03436v1-abstract-full').style.display = 'none'; document.getElementById('2303.03436v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">7 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2207.14120">arXiv:2207.14120</a> <span> [<a href="https://arxiv.org/pdf/2207.14120">pdf</a>, <a href="https://arxiv.org/ps/2207.14120">ps</a>, <a href="https://arxiv.org/format/2207.14120">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> </div> </div> <p class="title is-5 mathjax"> Relations among $\mathbb{P}$-Twists </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hochenegger%2C+A">Andreas Hochenegger</a>, <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2207.14120v2-abstract-short" style="display: inline;"> Given two $\mathbb{P}$-objects in some algebraic triangulated category, we investigate the possible relations among the associated $\mathbb{P}$-twists. The main result is that, under certain technical assumptions, the $\mathbb{P}$-twists commute if and only if the $\mathbb{P}$-objects are orthogonal. Otherwise, there are no relations at all. In particular, this applies to most of the known pairs o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.14120v2-abstract-full').style.display = 'inline'; document.getElementById('2207.14120v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2207.14120v2-abstract-full" style="display: none;"> Given two $\mathbb{P}$-objects in some algebraic triangulated category, we investigate the possible relations among the associated $\mathbb{P}$-twists. The main result is that, under certain technical assumptions, the $\mathbb{P}$-twists commute if and only if the $\mathbb{P}$-objects are orthogonal. Otherwise, there are no relations at all. In particular, this applies to most of the known pairs of $\mathbb{P}$-objects on hyperk盲hler varieties. In order to show this, we relate $\mathbb{P}$-twists to spherical twists and apply known results about the absence of relations between pairs of spherical twists. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.14120v2-abstract-full').style.display = 'none'; document.getElementById('2207.14120v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 July, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages, added references, minor improvements</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2206.11686">arXiv:2206.11686</a> <span> [<a href="https://arxiv.org/pdf/2206.11686">pdf</a>, <a href="https://arxiv.org/format/2206.11686">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Compactified Jacobians of Extended ADE Curves and Lagrangian Fibrations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Czapli%C5%84ski%2C+A">Adam Czapli艅ski</a>, <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a>, <a href="/search/math?searchtype=author&query=Lehn%2C+M">Manfred Lehn</a>, <a href="/search/math?searchtype=author&query=Rollenske%2C+S">S枚nke Rollenske</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2206.11686v2-abstract-short" style="display: inline;"> We observe that general reducible curves in sufficiently positive linear systems on K3 surfaces are of a form that generalises Kodaira's classification of singular elliptic fibres and thus call them extended ADE curves. On such a curve $C$, we describe a compactified Jacobian and show that its components reflect the intersection graph of $C$. This extends known results when $C$ is reduced, but n… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.11686v2-abstract-full').style.display = 'inline'; document.getElementById('2206.11686v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2206.11686v2-abstract-full" style="display: none;"> We observe that general reducible curves in sufficiently positive linear systems on K3 surfaces are of a form that generalises Kodaira's classification of singular elliptic fibres and thus call them extended ADE curves. On such a curve $C$, we describe a compactified Jacobian and show that its components reflect the intersection graph of $C$. This extends known results when $C$ is reduced, but new difficulties arise when $C$ is non-reduced. As an application, we get an explicit description of general singular fibres of certain Lagrangian fibrations of Beauville-Mukai type. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.11686v2-abstract-full').style.display = 'none'; document.getElementById('2206.11686v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 June, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Minor corrections. To appear in Commun. Contemp. Math</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2105.13740">arXiv:2105.13740</a> <span> [<a href="https://arxiv.org/pdf/2105.13740">pdf</a>, <a href="https://arxiv.org/ps/2105.13740">ps</a>, <a href="https://arxiv.org/format/2105.13740">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Extension groups of Tautological Bundles on Symmetric Products of Curves </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2105.13740v1-abstract-short" style="display: inline;"> We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if $E\neq \mathcal O_X$ is simple, then the natural map $\operatorname*{Ext}^1(E,E)\to \operatorname*{Ext}^1(E^{[n]},E^{[n]})$ is injective for every $n$. Along with previous results, this implies that $E\mapsto E^{[n]}$ defines an embedding of the mo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.13740v1-abstract-full').style.display = 'inline'; document.getElementById('2105.13740v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2105.13740v1-abstract-full" style="display: none;"> We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if $E\neq \mathcal O_X$ is simple, then the natural map $\operatorname*{Ext}^1(E,E)\to \operatorname*{Ext}^1(E^{[n]},E^{[n]})$ is injective for every $n$. Along with previous results, this implies that $E\mapsto E^{[n]}$ defines an embedding of the moduli space of stable bundles of slope $渭\notin[-1,n-1]$ on the curve $X$ into the moduli space of stable bundles on the symmetric product $X^{(n)}$. The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill--Noether loci of $X$ with the loci in the moduli space of stable bundles on $X^{(n)}$ where the dimension of the tangent space jumps. We also prove that $E^{[n]}$ is simple if $E$ is simple. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.13740v1-abstract-full').style.display = 'none'; document.getElementById('2105.13740v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2103.07787">arXiv:2103.07787</a> <span> [<a href="https://arxiv.org/pdf/2103.07787">pdf</a>, <a href="https://arxiv.org/ps/2103.07787">ps</a>, <a href="https://arxiv.org/format/2103.07787">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Discriminant of Tautological Bundles on Symmetric Products of Curves </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2103.07787v1-abstract-short" style="display: inline;"> We compute a formula for the discriminant of tautological bundles on symmetric powers of a complex smooth projective curve. It follows that the Bogomolov inequality does not give a new restriction to stability of these tautological bundles. It only rules out tautological bundles which are already known to have the structure sheaf as a destabilising subbundle. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.07787v1-abstract-full" style="display: none;"> We compute a formula for the discriminant of tautological bundles on symmetric powers of a complex smooth projective curve. It follows that the Bogomolov inequality does not give a new restriction to stability of these tautological bundles. It only rules out tautological bundles which are already known to have the structure sheaf as a destabilising subbundle. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.07787v1-abstract-full').style.display = 'none'; document.getElementById('2103.07787v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1909.04321">arXiv:1909.04321</a> <span> [<a href="https://arxiv.org/pdf/1909.04321">pdf</a>, <a href="https://arxiv.org/ps/1909.04321">ps</a>, <a href="https://arxiv.org/format/1909.04321">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1307/mmj/20216092">10.1307/mmj/20216092 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Derived categories of (nested) Hilbert schemes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Belmans%2C+P">Pieter Belmans</a>, <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1909.04321v2-abstract-short" style="display: inline;"> In this paper we provide several results regarding the structure of derived categories of (nested) Hilbert schemes of points. We show that the criteria of Krug-Sosna and Addington for the universal ideal sheaf functor to be fully faithful resp. a $\mathbb{P}$-functor are sharp. Then we show how to embed multiple copies of the derived category of the surface using these fully faithful functors. We… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.04321v2-abstract-full').style.display = 'inline'; document.getElementById('1909.04321v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1909.04321v2-abstract-full" style="display: none;"> In this paper we provide several results regarding the structure of derived categories of (nested) Hilbert schemes of points. We show that the criteria of Krug-Sosna and Addington for the universal ideal sheaf functor to be fully faithful resp. a $\mathbb{P}$-functor are sharp. Then we show how to embed multiple copies of the derived category of the surface using these fully faithful functors. We also give a semiorthogonal decomposition for the nested Hilbert scheme of points on a surface, and finally we give an elementary proof of a semiorthogonal decomposition due to Toda for the symmetric product of a curve. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.04321v2-abstract-full').style.display = 'none'; document.getElementById('1909.04321v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages, added reference to result in new version of Jiang--Leung preprint</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1903.01641">arXiv:1903.01641</a> <span> [<a href="https://arxiv.org/pdf/1903.01641">pdf</a>, <a href="https://arxiv.org/ps/1903.01641">ps</a>, <a href="https://arxiv.org/format/1903.01641">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.geomphys.2020.103597">10.1016/j.geomphys.2020.103597 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Fourier-Mukai transformation and logarithmic Higgs bundles on punctual Hilbert schemes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Biswas%2C+I">Indranil Biswas</a>, <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1903.01641v3-abstract-short" style="display: inline;"> Given a vector bundle $E$ on a smooth projective curve or surface $X$ carrying the structure of a $V$-twisted Hitchin pair for some vector bundle $V$, we observe that the associated tautological bundle $E^{[n]}$ on the punctual Hilbert scheme of points $X^{[n]}$ has an induced structure of a $((V^\vee)^{[n]})^\vee$-twisted Hitchin pair, where $(V^\vee)^{[n]}$ is a vector bundle on $X^{[n]}$ constr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.01641v3-abstract-full').style.display = 'inline'; document.getElementById('1903.01641v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1903.01641v3-abstract-full" style="display: none;"> Given a vector bundle $E$ on a smooth projective curve or surface $X$ carrying the structure of a $V$-twisted Hitchin pair for some vector bundle $V$, we observe that the associated tautological bundle $E^{[n]}$ on the punctual Hilbert scheme of points $X^{[n]}$ has an induced structure of a $((V^\vee)^{[n]})^\vee$-twisted Hitchin pair, where $(V^\vee)^{[n]}$ is a vector bundle on $X^{[n]}$ constructed using the dual $V^\vee$ of $V$. In particular, a Higgs bundle on $X$ induces a logarithmic Higgs bundle on the Hilbert scheme $X^{[n]}$. We then show that the known results on stability of tautological bundles and reconstruction from tautological bundles generalize to tautological Hitchin pairs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.01641v3-abstract-full').style.display = 'none'; document.getElementById('1903.01641v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version; Jour. Geom. Phys. (to appear)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14D23; 14D20; 14H30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1809.06450">arXiv:1809.06450</a> <span> [<a href="https://arxiv.org/pdf/1809.06450">pdf</a>, <a href="https://arxiv.org/ps/1809.06450">ps</a>, <a href="https://arxiv.org/format/1809.06450">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Stability of Tautological Bundles on Symmetric Products of Curves </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1809.06450v1-abstract-short" style="display: inline;"> We prove that, if $C$ is a smooth projective curve over the complex numbers, and $E$ is a stable vector bundle on $C$ whose slope does not lie in the interval $[-1,n-1]$, then the associated tautological bundle $E^{[n]}$ on the symmetric product $C^{(n)}$ is again stable. Also, if $E$ is semi-stable and its slope does not lie in the interval $(-1,n-1)$, then $E^{[n]}$ is semi-stable. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1809.06450v1-abstract-full" style="display: none;"> We prove that, if $C$ is a smooth projective curve over the complex numbers, and $E$ is a stable vector bundle on $C$ whose slope does not lie in the interval $[-1,n-1]$, then the associated tautological bundle $E^{[n]}$ on the symmetric product $C^{(n)}$ is again stable. Also, if $E$ is semi-stable and its slope does not lie in the interval $(-1,n-1)$, then $E^{[n]}$ is semi-stable. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1809.06450v1-abstract-full').style.display = 'none'; document.getElementById('1809.06450v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 September, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1808.05931">arXiv:1808.05931</a> <span> [<a href="https://arxiv.org/pdf/1808.05931">pdf</a>, <a href="https://arxiv.org/ps/1808.05931">ps</a>, <a href="https://arxiv.org/format/1808.05931">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Some ways to reconstruct a sheaf from its tautological image on a Hilbert scheme of points </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a>, <a href="/search/math?searchtype=author&query=Rennemo%2C+J+V">J酶rgen Vold Rennemo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1808.05931v2-abstract-short" style="display: inline;"> For $X$ a smooth quasi-projective variety and $X^{[n]}$ its associated Hilbert scheme of $n$ points, we study two canonical Fourier--Mukai transforms $D(X)\to D(X^{[n]})$, the one along the structure sheaf and the one along the ideal sheaf of the universal family. For $\dim X\ge 2$, we prove that both functors admit a left-inverse. This means in particular that both functors are faithful and injec… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.05931v2-abstract-full').style.display = 'inline'; document.getElementById('1808.05931v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1808.05931v2-abstract-full" style="display: none;"> For $X$ a smooth quasi-projective variety and $X^{[n]}$ its associated Hilbert scheme of $n$ points, we study two canonical Fourier--Mukai transforms $D(X)\to D(X^{[n]})$, the one along the structure sheaf and the one along the ideal sheaf of the universal family. For $\dim X\ge 2$, we prove that both functors admit a left-inverse. This means in particular that both functors are faithful and injective on isomorphism classes of objects. Using another method, we also show in the case of an elliptic curve that the Fourier--Mukai transform along the structure sheaf of the universal family is faithful and injective on isomorphism classes. Furthermore, we prove that the universal family of $X^{[n]}$ is always flat over $X$, which implies that the Fourier--Mukai transform along its structure sheaf maps coherent sheaves to coherent sheaves. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.05931v2-abstract-full').style.display = 'none'; document.getElementById('1808.05931v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 July, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 August, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Added J酶rgen Rennemo as co-author. Conjecture 1.6 in the previous version is now resolved, see Theorem 1.2 in this version</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1710.08618">arXiv:1710.08618</a> <span> [<a href="https://arxiv.org/pdf/1710.08618">pdf</a>, <a href="https://arxiv.org/ps/1710.08618">ps</a>, <a href="https://arxiv.org/format/1710.08618">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00029-021-00740-4">10.1007/s00029-021-00740-4 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Universal functors on symmetric quotient stacks of Abelian varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a>, <a href="/search/math?searchtype=author&query=Meachan%2C+C">Ciaran Meachan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1710.08618v2-abstract-short" style="display: inline;"> We consider certain universal functors on symmetric quotient stacks of Abelian varieties. In dimension two, we discover a family of $\mathbb{P}$-functors which induce new derived autoequivalences of Hilbert schemes of points on Abelian surfaces; a set of braid relations on a holomorphic symplectic sixfold; and a pair of spherical functors on the Hilbert square of an Abelian surface, whose twists a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1710.08618v2-abstract-full').style.display = 'inline'; document.getElementById('1710.08618v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1710.08618v2-abstract-full" style="display: none;"> We consider certain universal functors on symmetric quotient stacks of Abelian varieties. In dimension two, we discover a family of $\mathbb{P}$-functors which induce new derived autoequivalences of Hilbert schemes of points on Abelian surfaces; a set of braid relations on a holomorphic symplectic sixfold; and a pair of spherical functors on the Hilbert square of an Abelian surface, whose twists are related to the well-known Horja twist. In dimension one, our universal functors are fully faithful, giving rise to a semiorthogonal decomposition for the symmetric quotient stack of an elliptic curve (which we compare to the one discovered by Polishchuk--Van den Bergh), and they lift to spherical functors on the canonical cover, inducing twists which descend to give new derived autoequivalences here as well. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1710.08618v2-abstract-full').style.display = 'none'; document.getElementById('1710.08618v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 August, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 October, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Minor revisions. 35 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14D22; 14F05; 18E30 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Selecta Mathematica 2021 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1709.06434">arXiv:1709.06434</a> <span> [<a href="https://arxiv.org/pdf/1709.06434">pdf</a>, <a href="https://arxiv.org/ps/1709.06434">ps</a>, <a href="https://arxiv.org/format/1709.06434">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1112/S0010437X19007218">10.1112/S0010437X19007218 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Formality of $\mathbb{P}$-objects </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hochenegger%2C+A">Andreas Hochenegger</a>, <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1709.06434v2-abstract-short" style="display: inline;"> We show that a $\mathbb{P}$-object and simple configurations of $\mathbb{P}$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its gra… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.06434v2-abstract-full').style.display = 'inline'; document.getElementById('1709.06434v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.06434v2-abstract-full" style="display: none;"> We show that a $\mathbb{P}$-object and simple configurations of $\mathbb{P}$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its graded cohomology algebra. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.06434v2-abstract-full').style.display = 'none'; document.getElementById('1709.06434v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 May, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages, many changes to improve presentation, strengthened results in Section 5, same content as published version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 18E30; 14F05; 16E40 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Compositio Math. 155 (2019) 973-994 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1701.01331">arXiv:1701.01331</a> <span> [<a href="https://arxiv.org/pdf/1701.01331">pdf</a>, <a href="https://arxiv.org/ps/1701.01331">ps</a>, <a href="https://arxiv.org/format/1701.01331">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Derived categories of resolutions of cyclic quotient singularities </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a>, <a href="/search/math?searchtype=author&query=Ploog%2C+D">David Ploog</a>, <a href="/search/math?searchtype=author&query=Sosna%2C+P">Pawel Sosna</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1701.01331v2-abstract-short" style="display: inline;"> For a cyclic group $G$ acting on a smooth variety $X$ with only one character occurring in the $G$-equivariant decomposition of the normal bundle of the fixed point locus, we study the derived categories of the orbifold $[X/G]$ and the blow-up resolution $\widetilde Y \to X/G$. Some results generalise known facts about $X = A^n$ with diagonal $G$-action, while other results are new also in this… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1701.01331v2-abstract-full').style.display = 'inline'; document.getElementById('1701.01331v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1701.01331v2-abstract-full" style="display: none;"> For a cyclic group $G$ acting on a smooth variety $X$ with only one character occurring in the $G$-equivariant decomposition of the normal bundle of the fixed point locus, we study the derived categories of the orbifold $[X/G]$ and the blow-up resolution $\widetilde Y \to X/G$. Some results generalise known facts about $X = A^n$ with diagonal $G$-action, while other results are new also in this basic case. In particular, if the codimension of the fixed point locus equals $|G|$, we study the induced tensor products under the equivalence $D^b(\widetilde Y) \cong D^b([X/G])$ and give a 'flop-flop=twist' type formula. We also introduce candidates for general constructions of categorical crepant resolutions inside the derived category of a given geometric resolution of singularities and test these candidates on cyclic quotient singularities. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1701.01331v2-abstract-full').style.display = 'none'; document.getElementById('1701.01331v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 January, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages, many improvements from review, to appear in Quarterly J. Math</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14F05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1612.04348">arXiv:1612.04348</a> <span> [<a href="https://arxiv.org/pdf/1612.04348">pdf</a>, <a href="https://arxiv.org/ps/1612.04348">ps</a>, <a href="https://arxiv.org/format/1612.04348">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1612.04348v2-abstract-short" style="display: inline;"> We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs. As an application, we obtain shorter proofs for known results as well as new formu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.04348v2-abstract-full').style.display = 'inline'; document.getElementById('1612.04348v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1612.04348v2-abstract-full" style="display: none;"> We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs. As an application, we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves. In particular, we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.04348v2-abstract-full').style.display = 'none'; document.getElementById('1612.04348v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 January, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 December, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">References updated, some typos fixed</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1604.03537">arXiv:1604.03537</a> <span> [<a href="https://arxiv.org/pdf/1604.03537">pdf</a>, <a href="https://arxiv.org/ps/1604.03537">ps</a>, <a href="https://arxiv.org/format/1604.03537">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Varieties with $\mathbb P$-units </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1604.03537v1-abstract-short" style="display: inline;"> We study the class of compact K盲hler manifolds with trivial canonical bundle and the property that the cohomology of the trivial line bundle is generated by one element. If the square of the generator is zero, we get the class of strict Calabi--Yau manifolds. If the generator is of degree 2, we get the class of compact hyperk盲hler manifolds. We provide some examples and structure results for the c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1604.03537v1-abstract-full').style.display = 'inline'; document.getElementById('1604.03537v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1604.03537v1-abstract-full" style="display: none;"> We study the class of compact K盲hler manifolds with trivial canonical bundle and the property that the cohomology of the trivial line bundle is generated by one element. If the square of the generator is zero, we get the class of strict Calabi--Yau manifolds. If the generator is of degree 2, we get the class of compact hyperk盲hler manifolds. We provide some examples and structure results for the cases where the generator is of higher nilpotency index and degree. In particular, we show that varieties of this type are closely related to higher-dimensional Enriques varieties. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1604.03537v1-abstract-full').style.display = 'none'; document.getElementById('1604.03537v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 April, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2016. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1501.07253">arXiv:1501.07253</a> <span> [<a href="https://arxiv.org/pdf/1501.07253">pdf</a>, <a href="https://arxiv.org/ps/1501.07253">ps</a>, <a href="https://arxiv.org/format/1501.07253">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Symmetric quotient stacks and Heisenberg actions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1501.07253v1-abstract-short" style="display: inline;"> For every smooth projective variety, we construct an action of the Heisenberg algebra on the direct sum of the Grothendieck groups of all the symmetric quotient stacks which contains the Fock space as a subrepresentation. The action is induced by functors on the level of the derived categories which form a weak categorification of the action. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1501.07253v1-abstract-full" style="display: none;"> For every smooth projective variety, we construct an action of the Heisenberg algebra on the direct sum of the Grothendieck groups of all the symmetric quotient stacks which contains the Fock space as a subrepresentation. The action is induced by functors on the level of the derived categories which form a weak categorification of the action. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1501.07253v1-abstract-full').style.display = 'none'; document.getElementById('1501.07253v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 January, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1411.0824">arXiv:1411.0824</a> <span> [<a href="https://arxiv.org/pdf/1411.0824">pdf</a>, <a href="https://arxiv.org/ps/1411.0824">ps</a>, <a href="https://arxiv.org/format/1411.0824">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1112/jlms/jdv014">10.1112/jlms/jdv014 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Equivalences of equivariant derived categories </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a>, <a href="/search/math?searchtype=author&query=Sosna%2C+P">Pawel Sosna</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1411.0824v1-abstract-short" style="display: inline;"> We investigate conditions for a Fourier-Mukai transform between derived categories of coherent sheaves on smooth projective stacks endowed with actions by finite groups to lift to the associated equivariant derived categories. As an application we give a condition under which a global quotient stack cannot be derived equivalent to a variety. We also apply our techniques to generalised Kummer stack… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1411.0824v1-abstract-full').style.display = 'inline'; document.getElementById('1411.0824v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1411.0824v1-abstract-full" style="display: none;"> We investigate conditions for a Fourier-Mukai transform between derived categories of coherent sheaves on smooth projective stacks endowed with actions by finite groups to lift to the associated equivariant derived categories. As an application we give a condition under which a global quotient stack cannot be derived equivalent to a variety. We also apply our techniques to generalised Kummer stacks and symmetric quotients. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1411.0824v1-abstract-full').style.display = 'none'; document.getElementById('1411.0824v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 November, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">24 pages, comments welcome</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1405.1006">arXiv:1405.1006</a> <span> [<a href="https://arxiv.org/pdf/1405.1006">pdf</a>, <a href="https://arxiv.org/ps/1405.1006">ps</a>, <a href="https://arxiv.org/format/1405.1006">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> P-functor versions of the Nakajima operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1405.1006v1-abstract-short" style="display: inline;"> For every smooth quasi-projective surface X we construct a series of P^{n-1}-functors H_{l,n}: D(X x X^[l]) --> D(X^[n+l]) between the derived categories of the Hilbert schemes of points for n>max{l,1} using the derived McKay correspondence. They can be considered as analogues of the Nakajima operators. The functors also restrict to P^{n-1}-functors on the generalised Kummer varieties. We also stu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1405.1006v1-abstract-full').style.display = 'inline'; document.getElementById('1405.1006v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1405.1006v1-abstract-full" style="display: none;"> For every smooth quasi-projective surface X we construct a series of P^{n-1}-functors H_{l,n}: D(X x X^[l]) --> D(X^[n+l]) between the derived categories of the Hilbert schemes of points for n>max{l,1} using the derived McKay correspondence. They can be considered as analogues of the Nakajima operators. The functors also restrict to P^{n-1}-functors on the generalised Kummer varieties. We also study the induced autoequivalences and obtain, for example, a universal braid relation in the groups of derived autoequivalences of Hilbert squares of K3 surfaces and Kummer fourfolds. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1405.1006v1-abstract-full').style.display = 'none'; document.getElementById('1405.1006v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 May, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2014. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1404.2105">arXiv:1404.2105</a> <span> [<a href="https://arxiv.org/pdf/1404.2105">pdf</a>, <a href="https://arxiv.org/ps/1404.2105">ps</a>, <a href="https://arxiv.org/format/1404.2105">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> On the derived category of the Hilbert scheme of points on an Enriques surface </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a>, <a href="/search/math?searchtype=author&query=Sosna%2C+P">Pawel Sosna</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1404.2105v1-abstract-short" style="display: inline;"> We use semi-orthogonal decompositions to construct autoequivalences of Hilbert schemes of points on Enriques surfaces and of Calabi-Yau varieties which cover them. While doing this, we show that the derived category of a surface whose irregularity and geometric genus vanish embeds into the derived category of its Hilbert scheme of points. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1404.2105v1-abstract-full" style="display: none;"> We use semi-orthogonal decompositions to construct autoequivalences of Hilbert schemes of points on Enriques surfaces and of Calabi-Yau varieties which cover them. While doing this, we show that the derived category of a surface whose irregularity and geometric genus vanish embeds into the derived category of its Hilbert scheme of points. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1404.2105v1-abstract-full').style.display = 'none'; document.getElementById('1404.2105v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 April, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages, comments welcome</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1402.1651">arXiv:1402.1651</a> <span> [<a href="https://arxiv.org/pdf/1402.1651">pdf</a>, <a href="https://arxiv.org/ps/1402.1651">ps</a>, <a href="https://arxiv.org/format/1402.1651">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1215/00277630-2891370">10.1215/00277630-2891370 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Spherical functors on the Kummer surface </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a>, <a href="/search/math?searchtype=author&query=Meachan%2C+C">Ciaran Meachan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1402.1651v2-abstract-short" style="display: inline;"> We find two natural spherical functors associated to the Kummer surface and analyse how their induced twists fit with Bridgeland's conjecture on the derived autoequivalence group of a complex algebraic K3 surface. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1402.1651v2-abstract-full" style="display: none;"> We find two natural spherical functors associated to the Kummer surface and analyse how their induced twists fit with Bridgeland's conjecture on the derived autoequivalence group of a complex algebraic K3 surface. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1402.1651v2-abstract-full').style.display = 'none'; document.getElementById('1402.1651v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 September, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 February, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version. To appear in Nagoya Math. J</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14F05; 14J28; 18E30 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Nagoya Math. J., 219:1--8, 2015 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1301.4970">arXiv:1301.4970</a> <span> [<a href="https://arxiv.org/pdf/1301.4970">pdf</a>, <a href="https://arxiv.org/ps/1301.4970">ps</a>, <a href="https://arxiv.org/format/1301.4970">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> New derived autoequivalences of Hilbert schemes and generalised Kummer varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1301.4970v2-abstract-short" style="display: inline;"> We show that for every smooth projective surface X and every $n\ge 2$ the push-forward along the diagonal embedding gives a $P^{n-1}$-functor into the $S_n$-equivariant derived category of X^n. Using the Bridgeland--King--Reid--Haiman equivalence this yields a new autoequivalence of the derived category of the Hilbert scheme of n points on X. In the case that the canonical bundle of X is trivial a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1301.4970v2-abstract-full').style.display = 'inline'; document.getElementById('1301.4970v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1301.4970v2-abstract-full" style="display: none;"> We show that for every smooth projective surface X and every $n\ge 2$ the push-forward along the diagonal embedding gives a $P^{n-1}$-functor into the $S_n$-equivariant derived category of X^n. Using the Bridgeland--King--Reid--Haiman equivalence this yields a new autoequivalence of the derived category of the Hilbert scheme of n points on X. In the case that the canonical bundle of X is trivial and n=2 this autoequivalence coincides with the known EZ-spherical twist induced by the boundary of the Hilbert scheme. We also generalise the 16 spherical objects on the Kummer surface given by the exceptional curves to n^4 orthogonal $P^{n-1}$-Objects on the generalised Kummer variety. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1301.4970v2-abstract-full').style.display = 'none'; document.getElementById('1301.4970v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 May, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 January, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Mistakes in Lemma 3.3, Prop. 4.4, and Remark 4.7 corrected. The changes do not affect the main results</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1211.1640">arXiv:1211.1640</a> <span> [<a href="https://arxiv.org/pdf/1211.1640">pdf</a>, <a href="https://arxiv.org/ps/1211.1640">ps</a>, <a href="https://arxiv.org/format/1211.1640">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Tensor products of tautological bundles under the Bridgeland-King-Reid-Haiman equivalence </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1211.1640v1-abstract-short" style="display: inline;"> We give a description of the image of tensor products of tautological bundles on Hilbert schemes of points on surfaces under the Bridgeland-King-Reid-Haiman equivalence. Using this, some new formulas for cohomological invariants of these bundles are obtained. In particular, we give formulas for the Euler characteristic of arbitrary tensor products on the Hilbert scheme of two points and of triple… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1211.1640v1-abstract-full').style.display = 'inline'; document.getElementById('1211.1640v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1211.1640v1-abstract-full" style="display: none;"> We give a description of the image of tensor products of tautological bundles on Hilbert schemes of points on surfaces under the Bridgeland-King-Reid-Haiman equivalence. Using this, some new formulas for cohomological invariants of these bundles are obtained. In particular, we give formulas for the Euler characteristic of arbitrary tensor products on the Hilbert scheme of two points and of triple tensor products in general. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1211.1640v1-abstract-full').style.display = 'none'; document.getElementById('1211.1640v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 November, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2012. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1111.4263">arXiv:1111.4263</a> <span> [<a href="https://arxiv.org/pdf/1111.4263">pdf</a>, <a href="https://arxiv.org/ps/1111.4263">ps</a>, <a href="https://arxiv.org/format/1111.4263">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Extension groups of tautological sheaves on Hilbert schemes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1111.4263v2-abstract-short" style="display: inline;"> We give formulas for the extension groups between tautological sheaves and more general between tautological objects twisted by a determinant line bundle on the Hilbert scheme of points on a smooth quasi-projective surface. We do this using L. Scala's results about the image of tautological sheaves under the Bridgeland-King-Reid equivalence. We also compute the Yoneda products in terms of these fo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1111.4263v2-abstract-full').style.display = 'inline'; document.getElementById('1111.4263v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1111.4263v2-abstract-full" style="display: none;"> We give formulas for the extension groups between tautological sheaves and more general between tautological objects twisted by a determinant line bundle on the Hilbert scheme of points on a smooth quasi-projective surface. We do this using L. Scala's results about the image of tautological sheaves under the Bridgeland-King-Reid equivalence. We also compute the Yoneda products in terms of these formulas. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1111.4263v2-abstract-full').style.display = 'none'; document.getElementById('1111.4263v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 June, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 November, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Revised version</span> </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> 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