S[subscript 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[subscript n]-equivariant sheaf is equivalent to a resolution appearing in the theory of Koszul cohomology. Results: Our methods easily show that the dimension K[subscript p,q](B,L) is a polynomial in d for L=dA+P with A ample and d large enough. Conclusions: This interpretation allows us to extract various pieces of information about asymptotic properties Kp,q for fixed p,q.