Multigraded Castelnuovo-Mumford Regularity

By Diane Maclagan, Gregory G. Smith. 2003

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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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/39aa4cf2a255feb226b4f35867c7184ad89bc143

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Citations in the corpus

Multigraded Regularity: Syzygies and Fat Points (2004, background): “What extra conditions on M imply regB(M) ⊇ r(M)+N ? (Since the submission of this paper, Hà has given an example showing that this inclusion may not hold, as well as other related results in [15].”
Uniform bounds on multigraded regularity (2003, methodology): “The next section establishes our notation for toric varieties, recalls the definition of multigraded regularity from [ MS ] and collects the basic properties of multigraded Hilbert polynomials.”
Projective toric varieties as fine moduli spaces of quiver representations (2006, background): “, Lr−1; for the multigraded definition of regularity see [23], [16].”
Open Problems on Syzygies and Hilbert Functions (2009, background): “For Hilbert functions and regularity in that setting see [67, 87, 108, 114].”
Virtual resolutions for a product of projective spaces (2017, background): “etti numbers in a minimal free resolutions. However, on more general toric varieties, the multigraded Castelnuovo–Mumford regularity is not determined by a minimal free resolution; see Theorem 1.5 in [MS04] or Theorem 4.7 in [BC17]. From this perspective, we demonstrate that virtual resolutions impr”