Fano polytopes

Kasprzyk, Alexander; Nill, Benjamin

doi:10.1142/9789814412551_0017

Cited by 15 publications

(10 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[9]). For an overview of Fano polytopes, see [10]. Assumption (iii), although essential when considering qG-deformations of the toric variety X P , is not part of the definition of Fano polytope.…”

Section: (A) Fano Polytopesmentioning

confidence: 99%

Maximally mutable Laurent polynomials

et al. 2021

Self Cite

View full text Add to dashboard Cite

We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), which we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties with terminal locally toric singularities. We prove that there are exactly 10 mutation classes of rigid MMLPs in two variables; under mirror symmetry these correspond one-to-one with the 10 deformation classes of smooth del Pezzo surfaces. Furthermore, we give a computer-assisted classification of rigid MMLPs in three variables with reflexive Newton polytope; under mirror symmetry these correspond one-to-one with the 98 deformation classes of three-dimensional Fano manifolds with very ample anti-canonical bundle. We compare our proposal to previous approaches to constructing mirrors to Fano varieties, and explain why mirror symmetry in higher dimensions necessarily involves varieties with terminal singularities. Every known mirror to a Fano manifold, of any dimension, is a rigid MMLP.

show abstract

Section: (A) Fano Polytopesmentioning

confidence: 99%

Maximally mutable Laurent polynomials

et al. 2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…A Fano polytope P corresponds to a toric Fano variety X P via the spanning fan (that is, the fan whose cones are spanned by the faces of P). See [6] for the theory of toric varieties and [13] for a survey of Fano polytopes. When P is a Fano polygon, X P corresponds to a toric del Pezzo surface with at worst log terminal singularities.…”

Section: Example 25 (Mutation) Letmentioning

confidence: 99%

Quasi-period collapse for duals to Fano polygons: an explanation arising from algebraic geometry

Kasprzyk¹,

Wormleighton²

2018

Preprint

Self Cite

View full text Add to dashboard Cite

A. The Ehrhart quasi-polynomial of a rational polytope P is a fundamental invariant counting lattice points in integer dilates of P. The quasi-period of this quasi-polynomial divides the denominator of P but is not always equal to it: this is called quasi-period collapse. Polytopes experiencing quasi-period collapse appear widely across algebra and geometry, and yet the phenomenon remains largely mysterious. By using techniques from algebraic geometry -Q-Gorenstein deformations of orbifold del Pezzo surfaces -we explain quasi-period collapse for rational polygons dual to Fano polygons and describe explicitly the discrepancy between the quasi-period and the denominator. ILet P ⊂ Z d ⊗ Z Q be a convex lattice polytope of dimension d. Let L P (k) : kP ∩ Z d count the number of lattice points in dilations kP of P, k ∈ Z ≥0 . Ehrhart [9] showed that L P can be written as a degree dwhich we call the Ehrhart polynomial of P. The leading coefficient c d is given by Vol(P)/d!, c d−1 is equal to Vol(∂P)/2(d − 1)!, and c 0 1. Here Vol( • ) denotes the normalised volume, and ∂P denotes the boundary of P. For example, if P is two-dimensional (that is, P is a lattice polygon) we obtainSetting k 1 in this expression recovers Pick's Theorem [16]. The values of the Ehrhart polynomial of P form a generating function Ehr P (t) : k ≥0 L P (k)t k called the Ehrhart series of P. When the vertices of P are rational points the situation is more interesting. Recall that a quasi-polynomial with period s ∈ Z >0 is a function q : Z → Q defined by polynomials q 0 , q 1 , . . . , q s−1 such that q(k) q i (k) when k ≡ i (mod s).The degree of q is the largest degree of the q i . The minimum period of q is called the quasi-period, and necessarily divides any other period s. Ehrhart showed that L P is given by a quasi-polynomial of degree d, which we call the Ehrhart quasi-polynomial of P. Let π P denote the quasi-period of P. The smallest positive integer r P ∈ Z >0 such that r P P is a lattice polytope is called the denominator of P. It is certainly the case that L P is r P -periodic, however it is perhaps surprising that the quasi-period of L P does not always equal r P ; this phenomenon is called quasi-period collapse.Example 1.1 (Quasi-period collapse). Consider the triangle P : conv{(5, −1), (−1, −1), (−1, 1/2)} with denominator r P 2. This has L P (k) 9/2k 2 + 9/2k + 1, hence π P 1.Quasi-period collapse is poorly understood, although it occurs in many contexts. For example, de 8] consider polytopes arising naturally in the study of Lie algebras (the Gel'fand-Tsetlin polytopes and the polytopes determined by the Clebsch-Gordan coefficients) that exhibit quasiperiod collapse. In dimension two show that there exist rational polygons with r P arbitrarily large but with π P 1 (see also Example 3.8). give a constructive view of this phenomena in terms of GL d (Z)-scissor congruence; here a polytope is partitioned into pieces that are individually modified via GL d (Z) transformation and lattice translation, then reassembled to give a new polyt...

show abstract

“…They correspond to the smooth toric Fano varieties. For a summary of the various equivalences between lattice polytopes and toric Fano varieties, see [19].…”

Section: Introductionmentioning

confidence: 99%

Ehrhart Polynomial Roots of Reflexive Polytopes

Hegedüs

Higashitani

Kasprzyk

2019

Electron. J. Combin.

View full text Add to dashboard Cite

Recent work has focused on the roots z ∈ C of the Ehrhart polynomial of a lattice polytope P . The case when Re(z) = −1/2 is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when dim(P ) ≤ 7. We also consider the "half-strip condition", where all roots z satisfy − dim(P )/2 ≤ Re(z) ≤ dim(P )/2 − 1, and show that this holds for any reflexive polytope with dim(P ) ≤ 5. We give an example of a 10-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi-Shibata in dimension 34.2010 Mathematics Subject Classification. 52B20 (Primary); 05A15, 14M25 (Secondary).

show abstract

Fano polytopes

Cited by 15 publications

References 28 publications

Maximally mutable Laurent polynomials

Maximally mutable Laurent polynomials

Quasi-period collapse for duals to Fano polygons: an explanation arising from algebraic geometry

Ehrhart Polynomial Roots of Reflexive Polytopes

Contact Info

Product

Resources

About