95 research outputs found

    On the Hilbert scheme of degeneracy loci of twisted differential forms

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    We prove that, for 3 < m < n-1, the Grassmannian of m-dimensional subspaces of the space of skew-symmetric forms over a vector space of dimension n is birational to the Hilbert scheme of the degeneracy loci of m global sections of Omega(2), the twisted cotangent bundle on P^{n-1}. For 3=m<n-1 and n odd, this Grassmannian is proved to be birational to the set of Veronese surfaces parametrized by the Pfaffians of linear skew-symmetric matrices of order n.Comment: 19 pages. Minor corrections, exposition improved. To appear in Trans. Amer. Math. So

    Socle degrees for local cohomology modules of thickenings of maximal minors and sub-maximal Pfaffians

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    Let SS be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let II be the determinantal ideal of maximal minors or Pf\operatorname{Pf} the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu--Weyman--Witt to determine the SS-module structures of ExtSj(S/It,S)\operatorname{Ext}^j_S(S/I^t, S) and ExtSj(S/Pft,S)\operatorname{Ext}^j_S(S/\operatorname{Pf}^t, S), from which we get the degrees of generators of these Ext\operatorname{Ext} modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang on the socle degrees of local cohomology modules of the form Hmj(S/It)H^j_\mathfrak{m}(S/I^t).Comment: Final version. Comments welcome

    Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions

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    We present a systematic approach to studying the geometric aspects of Vinberg theta-representations. The main idea is to use the Borel-Weil construction for representations of reductive groups as sections of homogeneous bundles on homogeneous spaces, and then to study degeneracy loci of these vector bundles. Our main technical tool is to use free resolutions as an "enhanced" version of degeneracy loci formulas. We illustrate our approach on several examples and show how they are connected to moduli spaces of Abelian varieties. To make the article accessible to both algebraists and geometers, we also include background material on free resolutions and representation theory.Comment: 41 pages, uses tabmac.sty, Dedicated to David Eisenbud on the occasion of his 65th birthday; v2: fixed some typos and added reference
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