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On a conjecture on aCM and Ulrich sheaves on degeneracy loci

Vladimiro Benedetti, Fabio Tanturri

Abstract

In this paper we address a conjecture by Kleppe and Miró-Roig stating that suitable twists by line bundles (on the smooth locus) of the exterior powers of the normal sheaf of a standard determinantal locus are arithmetically Cohen--Macaulay, and even Ulrich when the locus is linear determinantal. We do so by providing a very simple locally free resolution of such sheaves obtained through the so-called Weyman's Geometric Method.

On a conjecture on aCM and Ulrich sheaves on degeneracy loci

Abstract

In this paper we address a conjecture by Kleppe and Miró-Roig stating that suitable twists by line bundles (on the smooth locus) of the exterior powers of the normal sheaf of a standard determinantal locus are arithmetically Cohen--Macaulay, and even Ulrich when the locus is linear determinantal. We do so by providing a very simple locally free resolution of such sheaves obtained through the so-called Weyman's Geometric Method.
Paper Structure (14 sections, 27 theorems, 71 equations)

This paper contains 14 sections, 27 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. A few preliminaries
  3. ACM and Ulrich sheaves
  4. Technical tools from representation theory
  5. Some multilinear algebra
  6. Some resolutions on determinantal varieties of minimal corank
  7. Kempf collapsings and the Geometric Method
  8. Pushforward of W_j
  9. The special case j=1
  10. Relativization of the resolutions to degeneracy loci
  11. On degeneracy loci of morphisms between vector bundles
  12. The resolutions pass to degeneracy loci
  13. Consequences and applications: degeneracy loci
  14. Consequences and applications: determinantal schemes

Key Result

Theorem 1

Let $X={\mathbb{P}}^n$, $r=\min(\operatorname{rank}(E),\operatorname{rank}(F))-1$ and $D_r(s)$ be of expected pure dimension. Then for any $j=0,\dots,\operatorname{codim}_X(D_r(s))$ there exist sheaves $\mathcal{W}_j$ which coincide on the smooth locus with $\wedge^j{\mathcal{N}}_{D_r(s)/ X}$ up to

Theorems & Definitions (55)

  • Theorem : Theorem A
  • Theorem : Theorem B
  • Theorem 2.1: ESW, Proposition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3: Bott--Borel--Weil Theorem
  • Remark 2.4: Exterior power of a tensor product
  • Remark 2.5: Tensor product by exterior power
  • Lemma 2.6
  • proof
  • ...and 45 more