Christian Haase

Lattice Polygons and the Number 2i + 7

0.1. How it all began. When the second author translated a result on algebraic sur faces into the language of lattice polygons using toric geometry, he got a simple inequality for lattice polygons. This inequality had originally been discovered by Scott [12]. The first author then found a third proof. Subsequently, both authors went through a phase of polygon addiction. Once you get started drawing lattice polygons on graph paper and discovering relations between their numerical invariants, it is not so easy to stop! (The gentle reader has been warned.) Thus, it was just unavoidable that the authors came up with new inequalities: Scotts inequality can be sharpened if one takes into account another invariant, which is de fined by peeling off the skins of the polygons like an onion (see Section 3).

Cayley decompositions of lattice polytopes and upper bounds for h*-polynomials

Abstract We give an effective upper bound on the h*-polynomial of a lattice polytope in terms of its degree and leading coefficient, confirming a conjecture of Batyrev. We deduce this bound as a consequence of a strong Cayley decomposition theorem which says, roughly speaking, that any lattice polytope with a large multiple that has no interior lattice points has a nontrivial decomposition as a Cayley sum of polytopes of smaller dimension. Polytopes with nontrivial Cayley decompositions correspond to projectivized sums of toric line bundles, and our approach is partially inspired by classification results of Fujita and others in algebraic geometry. In an appendix, we interpret our Cayley decomposition theorem in terms of adjunction theory for toric varieties.

On the Maximal Width of Empty Lattice Simplices

We construct d -dimensional empty lattice simplices of arbitrarily high volume from (d? 1)-dimensional ones, while preserving the lattice width. In particular, we give an example of infinitely many empty 4-simplices of width 2.

Polyhedral adjunction theory

In this paper we offer a combinatorial view on the adjunction theory of toric varieties. Inspired by classical adjunction theory of polarized algebraic varieties we explore two convex-geometric notions: the Q-codegree and the nef value of a rational polytope P. We prove a structure theorem for lattice polytopes P with large Q-codegree. For this, we define the adjoint polytope P-(s) as the set of those points in P whose lattice distance to every facet of P is at least s. It follows from our main result that if P-(s) is empty for some s < 2/(dim P + 2), then the lattice polytope P has lattice width one. This has consequences in Ehrhart theory and on polarized toric varieties with dual defect. Moreover, we illustrate how classification results in adjunction theory can be translated into new classification results for lattice polytopes.

Mini-Workshop: Projective Normality of Smooth Toric Varieties

The mini-workshop on ”Projective Normality of Smooth Toric Varieties” focused on the question of whether every projective embedding of a smooth toric variety is projectively normal. Equivalently, this question asks whether every lattice point in kP is the sum of k lattice points in P when P is a smooth (lattice) polytope. The workshop consisted of morning talks on different aspects of the problem, and afternoon discussion groups where participants from a variety of different backgrounds worked on specific examples and approaches. Mathematics Subject Classification (2000): 14M25, 52B20. Introduction by the Organisers The mini-workshop on Projective normality of smooth toric varieties, organized by Christian Haase (Berlin), Takayuki Hibi (Osaka), and Diane Maclagan (New Brunswick), was held August 12th-18th, 2007. A small group of researchers with backgrounds in combinatorics, commutative algebra, and algebraic geometry worked on the conjecture that embeddings of smooth toric varieties are projectively normal. This very basic question appears in different guises in algebraic geometry, commutative algebra, and integer programming, but specific cases also arise in additive number theory, representation theory, and statistics. See the summary by Diane Maclagan for three versions of the same question. There were a limited number of contributed talks in the mornings, setting the theme for the afternoon working groups. Monday morning began with Diane Maclagan describing the problem, and Winfried Bruns surveying the known results in the polyhedral formulation. This was followed on Tuesday morning by Benjamin J. Howard and Hidefumi Ohsugi on special cases of normality, and an 2284 Oberwolfach Report 39/2007 introductory talk by Milena Hering on the geometric vanishing theorem approach to the problem. On Wednesday morning Hal Schenck described a commutative algebra approach developed on site together with Greg Smith, while Sam Payne explained the Frobenius splitting approach. The commutative algebra approach, with optimization notes, continued in the talk of Ngô Viêt Trung on Thursday morning. Najmuddin Fakhruddin also explained his proof of the extended twodimensional case on Thursday morning. Finally, on Friday we heard from Christian Haase and Andreas Paffenholz on some techniques for showing normality in special cases, and Francisco Santos on lattice Delaunay simplices which are potential starting points in search for a counterexample. In the afternoons we split into working groups which then reported on their findings before dinner. These discussions continued through breaks, and in gaps between talks. The atmosphere of the group was very energetic, and we hope that the momentum generated during the meeting will continue with some of the ideas developed being pursued by the participants. As a direct outcome of the workshop, we would like to mention • many examples of very-ample-yet-non-normal polytopes found by Winfried Bruns, • a joint effort of Christian Haase, Benjamin Nill, Andreas Paffenholz, and Francisco Santos to (finally) settle the ample+nef additivity question in dimension two, as well as • a dynamic survey on projective normality and related questions to be edited by Diane Maclagan. The organizers and participants sincerely thank the institute for providing excellent working conditions and the unique Oberwolfach spirit. We are also grateful for funding from the NSF grant supporting young US-based participants, which allowed an extra participant to attend. In what follows we present, in addition to summaries of the talks, brief accounts on the outcome of brainstorming sessions and working groups. Christian Haase Takayuki Hibi Diane Maclagan Mini-Workshop: Projective Normality of Smooth Toric Varieties 2285 Mini-Workshop: Projective Normality of Smooth Toric Varieties

Finitely many smooth d-polytopes with N lattice points

We prove that for fixed n there are only finitely many embeddings of ℚ-factorial toric varieties X into ℙn that are induced by a complete linear system. The proof is based on a combinatorial result that implies that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with ≤ 12 lattice points.

Polytopes associated to dihedral groups

In this note we investigate the convex hull of those n ×  n permutation matrices that correspond to symmetries of a regular n -gon. We give the complete facet description. As an application, we show that this yields a Gorenstein polytope, and we determine the Ehrhart h * -vector.

Generating smooth lattice polytopes

A lattice polytope P is the convex hull of finitely many lattice points in Zd. It is smooth if each cone in the normal fan is unimodular. It has recently been shown that in fixed dimension the number of lattice equivalence classes of smooth lattice polytopes in dimension d with at most N lattice points is finite. We describe an algorithm to compute a representative in each equivalence class, and report on results in dimension 2 and 3 for N ≤ 12. Our algorithm is implemented as an extension to the software system polymake.

Convex-normal (Pairs of) Polytopes

In 2012 Gubeladze (Adv.\ Math.\ 2012) introduced the notion of k-convex-normal polytopes to show that integral polytopes all of whose edges are longer than 4d(d+1) have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no difference between k- and (k+1)-convex-normality (for k >= 3) and improve the bound to 2d(d+1). In the second part we extend the definition to pairs of polytopes and show that for rational polytopes P and Q, where the normal fan of P is a refinement of the normal fan of Q, if every edge e_P of P is at least d times as long as the corresponding edge e_Q of Q, then (P+Q) \cap \Z^d = (P\cap \Z^d) + (Q \cap \Z^d).

Lattices generated by skeletons of reflexive polytopes

Lattices generated by lattice points in skeletons of reflexive polytopes are essential in determining the fundamental group and integral cohomology of Calabi-Yau hypersurfaces. Here we prove that the lattice generated by all lattice points in a reflexive polytope is already generated by lattice points in codimension two faces. This answers a question of John Morgan.