On syzygies of Veronese embedding of arbitrary projective varieties

Park, Euisung

doi:10.1016/j.jalgebra.2009.03.022

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…Similar results have been obtained by Park [24] under some restrictive assumptions. In the last section we discuss multigraded variants of the results presented.…”

Section: Introductionsupporting

confidence: 88%

Koszul homology and syzygies of Veronese subalgebras

2010

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A graded K-algebra R has property Np if it is generated in degree 1, has\ud relations in degree 2 and the syzygies of order up to p on the relations are linear. The\ud Green–Lazarsfeld index of R is the largest p such that it satisfies the property Np.\ud Our main results assert that (under a mild assumption on the base field) the c-th\ud Veronese subring of a polynomial ring has Green–Lazarsfeld index c + 1. The\ud same conclusion also holds for an arbitrary standard graded algebra, provided C is very large

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“…Similar results have been obtained by Park [24] under some restrictive assumptions. In the last section we discuss multigraded variants of the results presented.…”

Section: Introductionsupporting

confidence: 88%

Koszul homology and syzygies of Veronese subalgebras

2010

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“…In view of the wide interest in syzygies of Veronese modules [5,14,19,27,30,31,32], the most interesting new graded Betti table that we obtain is that of X 6Σ ⊆ P 27 , i.e. the image of P 2 under the 6-uple embedding ν 6 , in characteristic 40 009.…”

Section: Introductionmentioning

confidence: 95%

Computing graded Betti tables of toric surfaces

Castryck

Cools

Demeyer

et al. 2019

Trans. Amer. Math. Soc.

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We present various facts on the graded Betti table of a projectively embedded toric surface, expressed in terms of the combinatorics of its defining lattice polygon. These facts include explicit formulas for a number of entries, as well as a lower bound on the length of the linear strand that we conjecture to be sharp (and prove to be so in several special cases). We also present an algorithm for determining the graded Betti table of a given toric surface by explicitly computing its Koszul cohomology, and report on an implementation in SageMath. It works well for ambient projective spaces of dimension up to roughly 25, depending on the concrete combinatorics, although the current implementation runs in finite characteristic only. As a main application we obtain the graded Betti table of the Veronese surface ν 6 (P 2 ) ⊆ P 27 in characteristic 40 009. This allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary Veronese surface ν d (P 2 ).

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“…However, such computations may be infeasible in practice even when the resolution of X is well understood. A theorem of Park [30] implies that if X is m-regular, then Z satisfies N 2,2k−m+1 for m 2 ⩽ k ⩽ m − 2 and N 2,k for k ⩾ m − 1. In particular we conclude that the Hankel index of the Veronese re-embeddings of X grow at least linearly, eventually.…”

Section: Applicationsmentioning

confidence: 99%

Do Sums of Squares Dream of Free Resolutions?

Blekherman¹,

Sinn²,

Velasco³

2017

SIAM J. Appl. Algebra Geometry

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Abstract. For a real projective variety X, the cone ΣX of sums of squares of linear forms plays a fundamental role in real algebraic geometry. The dual cone Σ * X is a spectrahedron, and we show that its convexity properties are closely related to homological properties of X. For instance, we show that all extreme rays of Σ * X have rank 1 if and only if X has Castelnuovo-Mumford regularity two. More generally, if Σ * X has an extreme ray of rank p > 1, then X does not satisfy the property N2,p. We show that the converse also holds in a wide variety of situations: the smallest p for which property N2,p does not hold is equal to the smallest rank of an extreme ray of Σ * X greater than one. We generalize the work of Blekherman, Smith, and Velasco on equality of nonnegative polynomials and sums of squares from irreducible varieties to reduced schemes and classify all spectrahedral cones with only rank 1 extreme rays. Our results have applications to the positive semidefinite matrix completion problem and to the truncated moment problem on projective varieties.Key words. sums of squares, spectrahedra, free resolutions, Castelnuovo-Mumford regularity AMS subject classifications. 14P05, 13D02, 52A99, 05C50 DOI. 10.1137/16M10845601. Introduction. Minimal free resolutions and spectrahedra are central objects of study in commutative algebra and convex geometry, respectively. We connect these disparate areas via real algebraic geometry and show a surprisingly strong connection between convexity properties of certain spectrahedra and the minimal free resolution of the defining ideal of the associated real variety. In the process, we address fundamental questions on the relationship between nonnegative polynomials and sums of squares.In real algebraic geometry, we associate two convex cones to a real projective variety X: the cone P X of quadratic forms that are nonnegative on X, and the cone Σ X of sums of squares of linear forms. A recent line of work shows that convexity properties of these cones are strongly related to geometric properties of the variety X over the complex numbers [2], [7], and [4]. We extend these novel connections into the realm of homological algebra by showing a direct link between the convex geometry of the dual convex cone Σ * X and property N 2,p of the defining ideal of X: for an integer p 1, the scheme X satisfies property N 2,p if the jth syzygy module of the homogeneous ideal of X is generated in degree j + 2 for all j < p.

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On syzygies of Veronese embedding of arbitrary projective varieties

Cited by 4 publications

References 14 publications

Koszul homology and syzygies of Veronese subalgebras

Koszul homology and syzygies of Veronese subalgebras

Computing graded Betti tables of toric surfaces

Do Sums of Squares Dream of Free Resolutions?

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