Multigraded Castelnuovo-Mumford Regularity

@article{Maclagan2003MultigradedCR,
  title={Multigraded Castelnuovo-Mumford Regularity},
  author={Diane Maclagan and Gregory G. Smith},
  journal={arXiv: Commutative Algebra},
  year={2003}
}
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… 

Figures from this paper

MULTIGRADED CASTELNUOVO-MUMFORD REGULARITY, a∗-INVARIANTS AND THE MINIMAL FREE RESOLUTION
In recent years, two different multigraded variants of CastelnuovoMumford regularity have been developed, namely multigraded regularity, defined by the vanishing of multigraded pieces of local
Characterizing Multigraded Regularity on Products of Projective Spaces
We explore the relationship between multigraded Castelnuovo–Mumford regularity, truncations, Betti numbers, and virtual resolutions. We prove that on a product of projective spaces X , the
Multigraded regularity, a∗-invariant and the minimal free resolution
In recent years, two different multigraded variants of Castelnuovo–Mumford regularity have been developed, namely multigraded regularity, defined by the vanishing of multigraded pieces of local
Castelnuovo Mumford Regularity with respect to multigraded ideals
Multigraded Regularity : Syzygies and Fat Points Jessica Sidman
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
Multigraded Regularity, a * -invariant and the Minimal Free Resolution
In recent years, two different multigraded variants of Castelnuovo-Mumford regularity have been developed, namely multigraded regularity, defined by the vanishing of multigraded pieces of local
Multigraded Regularity: Syzygies and Fat Points
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 connec-
Vanishing theorems and the multigraded regularity of nonsingular subvarieties
Given scheme-theoretic equations for a nonsingular subvariety, we prove that the higher cohomology groups for suitable twists of the corresponding ideal sheaf vanish. From this result, we obtain
Uniform bounds on multigraded regularity
We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we
...
...

References

SHOWING 1-10 OF 68 REFERENCES
Uniform bounds on multigraded regularity
We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we
A generalization of Castelnuovo regularity to Grassmann varieties
Abstract: Using the Beilinson–Kapranov spectral sequence, we define Castelnuovo regularity for a coherent sheaf on a Grassmann variety. We show that many formal properties of regularity over
Local Cohomology of Stanley–Reisner Rings with Supports in General Monomial Ideals☆
We study the local cohomology modules HiIΣ(k[Δ]) of the Stanley–Reisner ring k[Δ] of a simplicial complex Δ with support in the ideal IΣ ⊂ k[Δ] corresponding to a subcomplex Σ ⊂ Δ. We give a
Toric Hyperkahler Varieties
Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study
Multigraded Hilbert schemes
We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely
Dickson invariants, regularity and computation in group cohomology
In this paper, we investigate the commutative algebra of the cohomology ring $H^*(G,k)$ of a finite group $G$ over a field $k$. We relate the concept of quasi-regular sequence, introduced by Benson
Cohomology on Toric Varieties and Local Cohomology with Monomial Supports
TLDR
A grading on R which is coarser than the Z^n-grading such that each component of H^i_B(R) is finite dimensional is given and an effective way to compute these components is given.
Castelnuovo–Mumford regularity in biprojective spaces
We define the concept of regularity for bigraded modules over a bigraded polynomial ring. In this setting we prove analogs of some of the classical results on m-regularity for graded modules over
Power Sums, Gorenstein Algebras, and Determinantal Loci
Forms and catalecticant matrices.- Sums of powers of linear forms, and gorenstein algebras.- Tangent spaces to catalecticant schemes.- The locus PS(s, j r) of sums of powers, and determinantal loci
The homogeneous coordinate ring of a toric variety
This paper will introduce the homogeneous coordinate ring S of a toric variety X . The ring S is a polynomial ring with one variable for each one-dimensional cone in the fan ∆ determining X , and S
...
...