The regularity of points in multi-projective spaces

Harbourne, Brian; Tuyl, Adam Van

doi:10.1016/j.jpaa.2003.07.006

Cited by 17 publications

(8 citation statements)

References 13 publications

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“…The benefit of studying this class of ideals is that we already have some information on the regularity of these ideals (see [9,11,14] for more on this).…”

Section: Points In Multi-projective Spacesmentioning

confidence: 99%

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Multigraded regularity: Coarsenings and resolutions

2006

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Let S = k[x 1 , . . . , x n ] be a Z r -graded ring with deg(x i ) = a i ∈ Z r for each i and suppose that M is a finitely generated Z r -graded S-module. In this paper we describe how to find finite subsets of Z r containing the multidegrees of the minimal multigraded syzygies of M. To find such a set, we first coarsen the grading of M so that we can view M as a Z-graded S-module. We use a generalized notion of Castelnuovo-Mumford regularity, which was introduced by D. Maclagan and G. Smith, to associate to M a number which we call the regularity number of M. The minimal degrees of the multigraded minimal syzygies are bounded in terms of this invariant.

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“…The benefit of studying this class of ideals is that we already have some information on the regularity of these ideals (see [9,11,14] for more on this).…”

Section: Points In Multi-projective Spacesmentioning

confidence: 99%

“…For our first set of examples, we consider the ideals of sets of points in P n1 × • • • × P nr . The benefit of studying this class of ideals is that we already have some information on the regularity of these ideals (see [9,11,14] for more on this).…”

Section: 1mentioning

confidence: 99%

Multigraded regularity: Coarsenings and resolutions

2006

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“…In this situation there are several classifications. Giuffrida, Maggioni, and Ragusa [7], who helped to initiate the study of points in multiprojective spaces (see, for example [8,9,12,13,14,18,20,21,22] for more on these points), provided the first classification. They showed that ACM sets of points in P 1 × P 1 can be classified via their Hilbert functions.…”

Section: Introductionmentioning

confidence: 99%

ACM sets of points in multiprojective space

Guardo¹,

Tuyl

2008

Collect. Math.

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If X is a finite set of points in a multiprojective space P n 1 ×· · ·×P nr with r ≥ 2, then X may or may not be arithmetically Cohen-Macaulay (ACM). For sets of points in P 1 × P 1 there are several classifications of the ACM sets of points. In this paper we investigate the natural generalizations of these classifications to an arbitrary multiprojective space. We show that each classification for ACM points in P 1 × P 1 fails to extend to the general case. We also give some new necessary and sufficient conditions for a set of points to be ACM.

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“…[Hà07,SVTW06], try to unify these approaches and to generalize them [BC17]. When restricted to zero-dimensional closed subschemes of toric varieties, [HVT04,SVTW06] studies alternative definitions for fat points in multiprojective space and [S ¸S16] studies the regularity (in the sense of [MS03]) of homogeneous ideals generated by regular sequences over simplicial toric varieties.…”

Section: Regularity For Zero-dimensional Closed Subschemesmentioning

confidence: 99%

Toric Eigenvalue Methods for Solving Sparse Polynomial Systems

Bender¹,

Telen²

2020

Preprint

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We consider the problem of computing homogeneous coordinates of points in a zerodimensional subscheme of a compact toric variety X. Our starting point is a homogeneous ideal I in the Cox ring of X, which gives a global description of this subscheme. It was recently shown that eigenvalue methods for solving this problem lead to robust numerical algorithms for solving (nearly) degenerate sparse polynomial systems. In this work, we give a first description of this strategy for non-reduced, zero-dimensional subschemes of X. That is, we allow isolated points with arbitrary multiplicities. Additionally, we investigate the regularity of I to provide the first universal complexity bounds for the approach, as well as sharper bounds for weighted homogeneous, multihomogeneous and unmixed sparse systems, among others. We disprove a recent conjecture regarding the regularity and prove an alternative version. Our contributions are illustrated by several examples.

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The regularity of points in multi-projective spaces

Cited by 17 publications

References 13 publications

Multigraded regularity: Coarsenings and resolutions

Multigraded regularity: Coarsenings and resolutions

ACM sets of points in multiprojective space

Toric Eigenvalue Methods for Solving Sparse Polynomial Systems

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