No abstract

Abstract. We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomology. We establish the key properties of regularity: its connection with the minimal generators of a module and its behavior in exact sequences. For an ideal sheaf on a simplicial toric variety X, we prove that its multigraded regularity bounds the equations that cut out the associated subvariety. We also provide a criterion for testing if an ample line bundle on X gives a projectively normal embedding.

For a finite abelian group G ⊂ GL(n, k), we describe the coherent component Y θ of the moduli space M θ of θ-stable McKay quiver representations. This is a not-necessarily-normal toric variety that admits a projective birational morphism Y θ → A n k /G obtained by variation of Geometric Invariant Theory quotient. As a special case, this gives a new construction of Nakamura's G-Hilbert scheme Hilb G that avoids the (typically highly singular) Hilbert scheme of |G|-points in A n k . To conclude, we describe the toric fan of Y θ and hence calculate the quiver representation corresponding to any point of Y θ .

The Eisenbud-Green-Harris conjecture states that a homogeneous ideal in k[x 1 , . . . , x n ] containing a homogeneous regular sequence f 1 , . . . , f n with deg(f i ) = a i has the same Hilbert function as an ideal containing x ai i for 1 ≤ i ≤ n. In this paper we prove the Eisenbud-Green-Harris conjecture when a j > j−1 i=1 (a i − 1) for all j > 1. This result was independently obtained by the two authors.