95 research outputs found
On the Hilbert scheme of degeneracy loci of twisted differential forms
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
Let be the polynomial ring on the space of non-square generic matrices or
the space of odd-sized skew-symmetric matrices, and let be the
determinantal ideal of maximal minors or 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 -module structures of
and
, from which we get the
degrees of generators of these 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 .Comment: Final version. Comments welcome
Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions
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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