Counterexamples to the Eisenbud–Goto regularity conjecture

McCullough, Jason; Peeva, Irena

doi:10.1090/jams/891

Cited by 47 publications

(50 citation statements)

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“…, X n ] be a polynomial ring with the standard grading, and let I ⊆ S be a homogeneous prime ideal. Then Very recently, McCullough and Peeva [MP17] found counterexamples to this conjecture. However, we are going to show that a certain consequence of it is nevertheless true.…”

Section: Motivation From Unimodality Questionsmentioning

confidence: 98%

Ehrhart Theory of Spanning Lattice Polytopes

Hofscheier

Katthän

Nill

2017

International Mathematics Research Notices

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The key object in the Ehrhart theory of lattice polytopes is the numerator polynomial of the rational generating series of the Ehrhart polynomial, called h * -polynomial. In this paper we prove a new result on the vanishing of its coefficients. As a consequence, we get that h * i = 0 implies h * i+1 = 0 if the lattice points of the lattice polytope affinely span the ambient lattice. This generalizes a recent result in algebraic geometry due to Blekherman, Smith, and Velasco, and implies a polyhedral consequence of the Eisenbud-Goto conjecture. We also discuss how this study is motivated by unimodality questions and how it relates to decomposition results on lattice polytopes of given degree. The proof methods involve a novel combination of successive modifications of half-open triangulations and considerations of number-theoretic step functions.

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Section: Motivation From Unimodality Questionsmentioning

confidence: 98%

Ehrhart Theory of Spanning Lattice Polytopes

Hofscheier

Katthän

Nill

2017

International Mathematics Research Notices

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“…Equation (1) fails for arbitrary schemes, that is, when P is not prime. A surprising construction introduced by the second author and Peeva [16] produced the first examples of projective varieties failing this bound by producing from a known embedded scheme with large regularity, a new projective variety embedded in a much larger space which also has large regularity. This reinforces the need to control the singularities of X to ensure optimal estimates for its regularity; in particular, the Eisenbud-Goto conjecture remains open for arbitrary smooth projective varieties 2.…”

Section: Introductionmentioning

confidence: 99%

“…See also related work of Kwak-Park [12] and Noma [19]. There are also mild classes of singular surfaces for which Equation (1) holds, see [17].The process in [16] of constructing the examples of projective varieties failing Equation (1) involves two major steps. The first step is the construction of the Reeslike algebra, which defines a subvariety of a weighted projective space.…”

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confidence: 99%

“…Moreover, we show that, even though the Rees-like algebra is not Cohen-Macaulay when I is not principal, its canonical module is Cohen-Macaulay.Theorem C. (Theorem 6.11) Suppose k is a field with char(k) = 2, and S is a polynomial algebra over k. Let I an ideal with ht(I) ≥ 2. Then we can provide an explicit presentation matrix for the canonical module ω RL(I) of the Rees-like algebra RL(I), and an explicit description of its minimal free resolution.In particular, RL(I) is a small Cohen-Macaulay module.The interested reader may want to consult the statement and proof of Theorem 6.11 for the precise statement.While the varieties produced in [16] are highly singular, it is natural to consider the possibility of smooth hyperplane sections of those varieties. Using the above results and working over characteristic 0 fields, we exploit Bertini style arguments to show that the resulting smooth varieties satisfy Equation (1).…”

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confidence: 99%

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Singularities of Rees-like algebras

2020

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Recently, Peeva and the second author constructed irreducible projective varieties with regularity much larger than their degree, yielding counterexamples to the Eisenbud-Goto Conjecture. Their construction involved two new ideas: Rees-like algebras and step-by-step homogenization. Yet, all of these varieties are singular and the nature of the geometry of these projective varieties was left open. The purpose of this paper is to study the singularities inherent in this process. We compute the codimension of the singular locus of an arbitrary Rees-like algebra over a polynomial ring. We then show that the relative size of the singular locus can increase under step-by-step homogenization. To address this defect, we construct a new process, we call prime standardization, which plays a similar role as step-by-step homogenization but also preserves the codimension of the singular locus. This is derived from ideas of Ananyan and Hochster and we use this to study the regularity of certain smooth hyperplane sections of Rees-like algebras, showing that they all satisfy the Eisenbud-Goto Conjecture, as expected. On a more qualitative note, while Rees-like algebras are almost never Cohen-Macaulay and never normal, we characterize when they are seminormal, weakly normal, and, in positive characteristic, F-split. Finally, we construct a finite free resolution of the canonical module of a Rees-like Algebra over the presenting polynomial ring showing that it is always Cohen-Macaulay and has a surprising self-dual structure.2010 Mathematics Subject Classification. 13D02,14B05. 1 2 P. MANTERO, J. MCCULLOUGH, AND L. E. MILLER or even some mildly singular varieties. There are many cases where the conjecture does hold including the case of curves [8] and smooth surfaces in characteristic 0 [21,13], and certain 3-folds in characteristic 0 [22]. See also related work of Kwak-Park [12] and Noma [19]. There are also mild classes of singular surfaces for which Equation (1) holds, see [17].The process in [16] of constructing the examples of projective varieties failing Equation (1) involves two major steps. The first step is the construction of the Reeslike algebra, which defines a subvariety of a weighted projective space. Specifically, given a homogeneous ideal I in a polynomial ring S over a field k, the Reeslike algebra of I is the non-standard graded k-algebra RL(The second step, which applies to any homogeneous ideal in a non-standard graded polynomial ring, produces an associated ideal in a much larger polynomial ring called its step-by-step homogenization. Unlike the usual homogenization of an ideal which defines the projective closure of an affine variety, the step-by-step homogenization produces a much larger variety; however, it preserves graded Betti numbers and primeness for nondegenerate primes, making it sufficient to produce the counterexamples to Equation (1).Thus far, explicit understanding of the geometry of the processes involved in both of these two steps is lacking. It was proved in [16] that Rees-like algebras are no...

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“…Let G be a standard graded algebra over a field k. It is an important problem in commutative algebra and algebraic geometry to find formulas and inequalities that relate the multiplicity or degree e(G) to other invariants of G, such as the codimension, degrees of the defining equations, or degrees of the higher syzygies. Significant advancements have been achieved in this area in recent years, see for instance [4,6,17,26].…”

Section: Introductionmentioning

confidence: 99%