Weighted Ehrhart theory and orbifold cohomology

Stapledon, Alan

doi:10.1016/j.aim.2008.04.010

Cited by 38 publications

(62 citation statements)

References 23 publications

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“…When s = d,δ P (t) = δ P (t) and the theorem below is due to Betke and McMullen (Theorem 5 in [4]). This case was also proved in Remark 3.5 in [21].…”

Section: Lemma 212 With the Notation Abovementioning

confidence: 65%

Inequalities and Ehrhart $\delta $-vectors

Stapledon

2009

Trans. Amer. Math. Soc.

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Abstract. For any lattice polytope P , we consider an associated polynomial δ P (t) and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known inequalities satisfied by the coefficients of the Ehrhart δ-vector of a lattice polytope. We also provide combinatorial proofs of two results of Stanley that were previously established using techniques from commutative algebra. Finally, we give a necessary numerical criterion for the existence of a regular unimodular lattice triangulation of the boundary of a lattice polytope.

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“…When s = d,δ P (t) = δ P (t) and the theorem below is due to Betke and McMullen (Theorem 5 in [4]). This case was also proved in Remark 3.5 in [21].…”

Section: Lemma 212 With the Notation Abovementioning

confidence: 65%

Inequalities and Ehrhart $\delta $-vectors

Stapledon

2009

Trans. Amer. Math. Soc.

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“…We give three methods in order to compute Spec geo P , showing that it yields finally a spectrum of P in the sense of Section 3. The first one and the third one are inspired by the works of Mustaţȃ-Payne [21] and Stapledon [25]. The second one is inspired by Batyrev's stringy E-functions.…”

Section: Various Interpretationsmentioning

confidence: 99%

“…• to construct a stacky version of the E-polynomial, the geometric spectrum of P : we define Spec geo P (z) := (z − 1) n v∈N z −ν(v) where ν is the Newton function of the polytope P , see Section 4. This geometric spectrum is closely related to the Ehrhart series and to the δvector of the polytope P , more precisely to their twisted versions studied by Stapledon [25] and Mustaţȃ-Payne [21]; it is also an orbifold Poincaré series (see Corollary 4.5), thanks to the description of the orbifold cohomology given by Borisov, Chen and Smith [5,Proposition 4.7],…”

Section: Introductionmentioning

confidence: 99%

Global spectra, polytopes and stacky invariants

Douai

2017

Math. Z.

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Given a convex polytope, we define its geometric spectrum, a stacky version of Batyrev's stringy E-functions, and we prove a stacky version of a formula of Libgober and Wood about the E-polynomial of a smooth projective variety. As an application, we get a closed formula for the variance of the geometric spectrum and a Noether's formula for two dimensional Fano polytopes (polytopes whose vertices are primitive lattice points; a Fano polytope is not necessarily smooth). We also show that this geometric spectrum is equal to the algebraic spectrum (the spectrum at infinity of a tame Laurent polynomial whose Newton polytope is the polytope alluded to). This gives an explanation and some positive answers to Hertling's conjecture about the variance of the spectrum of tame regular functions. * Partially supported by the grant ANR-13-IS01-0001-01 of the Agence nationale de la recherche. Mathematics Subject Classification 32S40, 14J33, 34M35, 14C40. Key words and phrases: Mirror symmetry, toric varieties, polytopes, orbifold cohomology, spectrum of regular tame functions. Y , with primitive generators v 1 , · · · , v r and associated divisors D 1 , · · · , D r . Put, for a subset J ⊂ I := {1, · · · , r}, D J := ∩ j∈J D j if J = ∅ and D J := Y if J = ∅ and define E st,P (z) := J⊂I

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“…The proof should be compared with the combinatorial proof of Theorem 1.2 (see [32]) and the proof of [26, Theorem 1.1]. Recall that σ denotes the cone over P B in N R × R n and that Σ H is a fan refining σ, such that the cones σ C in Σ H that do not lie in the boundary of σ are in bijection with the bounded cells C of H. The cone σ C has codimension equal to dim C, and its rays correspond to the primitive integer vectors…”

Section: Ehrhart Theory For Lawrence Polytopesmentioning

confidence: 99%

“…Ehrhart δ-polynomials of lattice polytopes have been studied extensively over the last forty years by many authors, including Stanley [28,29,30, 31] and Hibi [16,17,18,19]. In a recent paper [32], the author expressed the coefficients of the Ehrhart δ-polynomial of a lattice polytope as sums of dimensions of orbifold cohomology groups of a toric orbifold. We will use the following special case (cf.…”

Section: Introductionmentioning

confidence: 99%

Ehrhart theory for Lawrence polytopes and orbifold cohomology of hypertoric varieties

Stapledon

2009

Proc. Amer. Math. Soc.

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Abstract. We establish a connection between the orbifold cohomology of hypertoric varieties and the Ehrhart theory of Lawrence polytopes. More specifically, we show that the dimensions of the orbifold cohomology groups of a hypertoric variety are equal to the coefficients of the Ehrhart δ-polynomial of the associated Lawrence polytope. As a consequence, we deduce a formula for the Ehrhart δ-polynomial of a Lawrence polytope and use the injective part of the Hard Lefschetz Theorem for hypertoric varieties to deduce some inequalities between the coefficients of the δ-polynomial.

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Weighted Ehrhart theory and orbifold cohomology

Cited by 38 publications

References 23 publications

Inequalities and Ehrhart $\delta $-vectors

Inequalities and Ehrhart $\delta $-vectors

Global spectra, polytopes and stacky invariants

Ehrhart theory for Lawrence polytopes and orbifold cohomology of hypertoric varieties

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