Cayley-Bacharach Schemes and Their Canonical Modules

Geramita, Anthony V.; Kreuzer, Martin; Robbiano, Lorenzo

doi:10.2307/2154213

Cited by 43 publications

(61 citation statements)

References 2 publications

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“…Then e n i=1 a i − δ n i=b+1 a i . Conjecture 1.2 has been studied by several researchers, from very different points of view; for instance, see [GKR93,GK13,CDS20a,HLU20]. The validity of the EGH Conjecture in the case r = n for almost complete intersections would imply Conjecture 1.2.…”

mentioning

confidence: 99%

Some cases of the Eisenbud-Green-Harris conjecture

Caviglia

Maclagan

2008

Mathematical Research Letters

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The Eisenbud-Green-Harris conjecture states that a homogeneous ideal in k[x 1 , . . . , x n ] containing a homogeneous regular sequence f 1 , . . . , f n with deg(f i ) = a i has the same Hilbert function as an ideal containing x ai i for 1 ≤ i ≤ n. In this paper we prove the Eisenbud-Green-Harris conjecture when a j > j−1 i=1 (a i − 1) for all j > 1. This result was independently obtained by the two authors.

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confidence: 99%

Some cases of the Eisenbud-Green-Harris conjecture

Caviglia

Maclagan

2008

Mathematical Research Letters

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“…Recall that X is Cayley-Bacharach if the Hilbert function of X \ {P } does not depend on the point P ∈ X. Since this property is equivalent to the faithfulness of [ω] −a R (see [8]), the above characterization in terms of the core then follows as an immediate consequence of Corollary 4.5. We also show that if a large enough subset of X lies on a hypersurface of low degree, then the initial degree of core(m) is forced to be unexpectedly small (see Corollary 5.4 and Proposition 6.1), underlining once more the fact that the shape of the core reflects uniformity properties of the set of points, or the lack thereof.…”

Section: Introductionmentioning

confidence: 94%

Untitled

2010

Trans. Amer. Math. Soc.

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We relate the annihilators of graded components of the canonical module of a graded Cohen-Macaulay ring to colon ideals of powers of the homogeneous maximal ideal. In particular, we connect them to the core of the maximal ideal. An application of our results characterizes Cayley-Bacharach sets of points in terms of the structure of the core of the maximal ideal of their homogeneous coordinate ring. In particular, we show that a scheme is Cayley-Bacharach if and only if the core is a power of the maximal ideal.Thus the main goal of this paper is to study, quite broadly, the interplay between annihilators of graded components of ω on the one hand and colon ideals of powers of m on the other hand.More generally, let R be a standard graded Cohen-Macaulay algebra over a field k, of dimension d > 0, and write m for its homogeneous maximal ideal and ω = ω R for its graded canonical module. By a = a(R) we denote the a-invariant of R, which is the negative of the initial degree of ω. Recall that if J is any ideal generated by a linear system of parameters, then 'J is a minimal reduction of m with reduction number r J (m) = a + d ', which simply means that m i = J i−j m j for every i ≥ j ≥ a + d, but for no smaller j. It is easy to see that J i : m j = m i−j+a+d for every i and j ≥ a + d, provided R is Gorenstein or, more generally, level, which means that ω is generated by homogeneous elements of the same degree, ω = [ω] −a R. Under these assumptions, the R-submodules [ω] t R of ω generated by the homogeneous elements of a fixed degree t are all faithful as long as t ≥ −a; this is obvious since ω = [ω] −a R is faithful and there exists a form of positive degree regular on ω. The same holds if R is a domain because a suitable shift of ω embeds into R. Without the additional assumptions on the ring, neither the statement about the colon ideals nor the one about the graded components of the canonical module are true, but there is a close relationship between the two conditions. In fact in one of our main results we express, more generally, the annihilators of graded components of ω in terms of colon ideals of powers of m,for every t and j ≥ a + d (see Theorem 2.7). Conversely, this allows one to write the colon ideals J i : m j for any i and j ≥ a + d in the formwhere m i−j+a+d is the 'expected' part and N is an ideal of height zero that is generated in degrees ≤ i − j + a + d − 1 and can be described as(see Corollaries 2.14 and 2.15). To prove these results it is useful to replace the colon ideals J i : R m j in R by the corresponding colon ideals J i ω : ω m j in ω, which we then relate to truncations [ω] ≥i−j+d of ω and to graded components of the canonical module Ω of the extended Rees ring of m. This is done in one of our main technical results. There we also use a bound on the regularity of Ω to show that Theorem 2.3 and, for related results, [10,14,11,4]).Our results imply, for instance, that [ω] −a R is faithful, equivalently, [ω] t R is faithful for every t ≥ −a, if and only if J i : m j = m i−j+a+d for every i and ...

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“…As S/I(X) is a 1-dimensional Cohen-Macaulay graded algebra [8], the regularity index of S/I(X) is the Castelnuovo-Mumford regularity of S/I(X) [6]. By Hilbert-Serre Theorem, the Hilbert series of S/I(X) can be uniquely written as F X (t) = f (t) 1−t , where f is a polinomial of degree equal to the regularity of S/I(X).…”

Section: Preliminaries: Codes Associated To a Graphsmentioning

confidence: 99%

Vanishing Ideals Over Odd Cycles

Uribe-Paczka

Sarmiento

Márquez

2019

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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Let K be a finite field. Let X* be a subset of the a ne space Kn, which is parameterized by odd cycles. In this paper we give an explicit Gröbner basis for the vanishing ideal, I(X*), of X*. We give an explicit formula for the regularity of I(X*) and finally if X* is parameterized by an odd cycle of length k, we show that the Hilbert function of the vanishing ideal of X* can be written as linear combination of Hilbert functions of degenerate torus.

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Cayley-Bacharach Schemes and Their Canonical Modules

Cited by 43 publications

References 2 publications

Some cases of the Eisenbud-Green-Harris conjecture

Some cases of the Eisenbud-Green-Harris conjecture

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Vanishing Ideals Over Odd Cycles

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