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 m...

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is Z[x, y], which establishes a conjecture of Haiman and Sturmfels. © 2009 Gregory G. Smith. Published by Elsevier Inc. All rights reserved.

Let R = k[x1, . . . , xn] be a polynomial ring over a field k. We present a characterization of multigraded R-modules in terms of the minors of their presentation matrix. We describe explicitly the second syzygies of any multigraded R-module.

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is Z[x, y], which establishes a conjecture of Haiman and Sturmfels.

After [4] the shellability of multicomplexes Γ is given in terms of some special faces of Γ called facets. Here we give a criterion for the shellability in terms of maximal facets. Multigraded pretty clean filtration is the algebraic counterpart of a shellable multicomplex. We give also a criterion for the existence of a multigraded pretty clean filtration.

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 cohomol-ogy. We establish the key properties of regularity: its connection with the ...

The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P n1 × · · · × P k . Introduction L...

Let S be a finitely generated standard multigraded algebra over an Artinian local ring A; M a finitely generated multigraded S-module. This paper answers to the question when mixed multiplicities of M are positive and characterizes them in terms of lengths of A-modules. As an application, we get interesting results on mixed multiplicities of ideals, and recover some early results in [Te] and [TV].