S n -equivariant sheaves and Koszul cohomology

Abstract

Purpose: We give a new interpretation of Koszul cohomology, which is equivalent under the Bridgeland-King-Reid equivalence to Voisin's Hilbert scheme interpretation in dimensions 1 and 2 but is different in higher dimensions. Methods: We show that an explicit resolution of a certain S n -equivariant sheaf is equivalent to a resolution appearing in the theory of Koszul cohomology. Results: Our methods easily show that the dimension

A vanishing theorem for weight-one syzygies

“…Unfortunately we do now know whether such a construction is possible. Instead, what we do in effect is to use the ideas of the third author from [22] to construct a map α B , and show that the non-vanishing of K p,1 is implied by the non-vanishing of a certain quotient sheaf of A 2 . We show that a reduced (p + 1)-cycle that fails to impose independent conditions on H 0 (X, B) must appear in the support of this quotient, and this leads to the stated non-vanishing.…”

“…We start by recalling the results [22] of the third author interpreting K p,1 as an equivariant cohomology group. Consider then a very ample line bundle L on the smooth complex projective variety X.…”

Untitled

“…To prove a nonvanishing statement such as Theorem B, one needs a geometric interpretation of K p,1 itself. Voisin [2002; achieves this by working on a Hilbert scheme -which has the advantage of being smooth when dim X = 2 -while Yang [2014] passes in effect to the symmetric product. 3 We follow the latter approach for Theorem B: we exhibit a sheaf on Sym p+1 (X ) whose twisted global sections compute K p,1 (X, B; L d ), and we show that it is nonzero provided that there is a reduced cycle that fails to impose independent conditions on H 0 (X, B).…”

Syzygies of projective varieties of large degree: Recent progress and open problems

“…Inspired by Yang's interpretation of Koszul cohomology in [37], Yang and the authors establish in [11] the following: Conversely, if there is a reduced zero cycle w = x 1 + . .…”