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Gorenstein Toric Fano Varieties

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Gorenstein Toric Fano Varieties Gorenstein toric Fano varieties Benjamin Nill Dissertation der FakultÄat furÄ Mathematik und Physik der Eberhard-Karls-UniversitÄat TubingenÄ zur Erlangung des Grades eines Doktors der Naturwissenschaften vorgelegt 2005 Tag der mundlicÄ hen Quali¯kation: 22.07.2005 Dekan: Prof. Dr. P. Schmid 1. Berichterstatter: Prof. Dr. V. Batyrev 2. Berichterstatter: Prof. Dr. J. Hausen Gorenstein toric Fano varieties Benjamin Nill (Tubingen)Ä Dissertation der FakultÄat furÄ Mathematik und Physik der Eberhard-Karls-UniversitÄat TubingenÄ zur Erlangung des Grades eines Doktors der Naturwissenschaften vorgelegt 2005 FurÄ Jule Contents Introduction 9 Notation 15 1 Fans, polytopes and toric varieties 19 1.1 Cones and fans . 19 1.2 The classical construction of a toric variety from a fan . 20 1.3 The category of toric varieties . 20 1.4 The class group, the Picard group and the Mori cone . 21 1.5 Polytopes and lattice points . 23 1.6 Big and nef Cartier divisors . 26 1.7 Ample Cartier divisors and projective toric varieties . 28 2 Singularities and toric Fano varieties 31 2.1 Resolution of singularities and discrepancy . 31 2.2 Singularities on toric varieties . 35 2.3 Toric Fano varieties . 36 3 Reexive polytopes 41 3.1 Basic properties . 43 3.2 Projecting along lattice points on the boundary . 46 3.3 Pairs of lattice points on the boundary . 52 3.4 Classi¯cation results in low dimensions . 55 3.5 Sharp bounds on the number of vertices . 58 3.6 Reexive simplices . 68 3.6.1 Weight systems of simplices . 68 3.6.2 The main result . 73 3.7 Lattice points in reexive polytopes . 81 3.7.1 The Ehrhart polynomial . 81 3.7.2 Bounds on the volume and lattice points . 83 3.7.3 Counting lattice points in residue classes . 88 4 Terminal Gorenstein toric Fano 3-folds 89 4.1 Primitive collections and relations . 90 4.2 Combinatorics of quasi-smooth Fano polytopes . 94 4.2.1 De¯nition and basic properties . 94 4.2.2 Projections of quasi-smooth Fano polytopes . 97 7 8 Contents 4.3 Classi¯cation of quasi-smooth Fano polytopes . 101 4.3.1 The main theorem . 101 4.3.2 Classi¯cation when no AS-points exist . 102 4.3.3 Classi¯cation when AS-points exist . 104 4.4 Table of quasi-smooth Fano polytopes . 113 5 The set of roots 119 5.1 The set of roots of a complete toric variety . 121 5.2 The set of roots of a reexive polytope . 130 5.3 Criteria for a reductive automorphism group . 133 5.4 Symmetric toric varieties . 138 5.5 Successive sums of lattice points . 140 5.6 Examples . 143 6 Centrally symmetric reexive polytopes 147 6.1 Roots . 148 6.2 Vertices . 149 6.3 Classi¯cation theorem . 151 6.4 Embedding theorems . 157 6.5 Lattice points . 160 Index 165 Bibliography 168 Appendix A - Zusammenfassung in deutscher Sprache 175 Appendix B - Lebenslauf 181 Introduction In this thesis we concern ourselves with Gorenstein toric Fano varieties, that is, with complete normal toric varieties whose anticanonical divisor is an am- ple Cartier divisor. These algebraic-geometric objects correspond to reexive polytopes introduced by Batyrev in [Bat94]. Reexive polytopes are lattice polytopes containing the origin in their interior such that the dual polytope also is a lattice polytope. It was shown by Batyrev that the associated vari- eties are ambient spaces of Calabi-Yau hypersurfaces and together with their duals naturally yield candidates for mirror symmetry pairs. This has raised a lot of interest in this special class of lattice polytopes among physicists and mathematicians. It is known that in ¯xed dimension d there only are a ¯nite number of isomorphism classes of d-dimensional reexive polytopes. Using their computer program PALP [KS04a] Kreuzer and Skarke succeeded in classifying d-dimensional reexive polytopes for d 4 [KS98, KS00, KS04b]. They found 16 isomorphism classes for d = 2, 4319 for· d = 3, and 473800776 for d = 4. While there are many papers devoted to the study and classi¯cation of non- singular toric Fano varieties [WW82, Bat82a, Bat82b, Bat99, Sat00, Deb03, Cas03a, Cas03b], in the singular case there has not yet been done so much, es- pecially in higher dimensions. This can be explained by several di±culties: First many algebraic-geometric methods like birational factorization, Riemann-Roch or intersection theory cannot simply be applied, especially since there need not exist a crepant toric resolution. Second most convex-geometric proofs relied on the vertices of a facet forming a lattice basis, a fact which is no longer true for reexive polytopes, where facets can even contain lattice points in their in- terior. Third the huge number of reexive polytopes causes any classi¯cation approach to depend heavily on computer calculations, hence often we do not get mathematically satisfying proofs even when restricting to low dimensions. The aim of this thesis is to give a ¯rst systematic mathematical investiga- tion of Gorenstein toric Fano varieties by thorougly examining the combinatorial and geometric properties of their convex-geometric counterparts, that is, reex- ive polytopes. We would like to generalize useful tools and theorems previously only known to hold for nonsingular toric Fano varieties to the case of mild singu- larities and to prove classi¯cation theorems in important cases and in arbitrary dimension. Moreover we are interested in ¯nding constraints on the combina- torics of reexive polytopes and conjectures and sharp bounds on invariants that can explain interesting observations made in the large computer data. As we will see most of these aims are actually not out of reach, for instance it will unexpectedly turn out that Q-factorial Gorenstein toric Fano varieties are in many aspects nearly as benign as nonsingular toric Fano varieties. More- over by these generalizations with a strong focus on combinatorics even results previously already proven in the nonsingular case become more transparent. 9 10 Introduction Above all this work provides useful tools, several conjectures to prove in the future and many results that show reexive polytopes to be truly interesting objects - not only from a physicist's but also from a pure mathematician's point of view. This thesis is organized in six chapters. Any major chapter (3-6) starts with an introductory section, in which also an explicit list of the most important new results is contained. Furthermore the reader will ¯nd right after this introduc- tion a summary of notation and at the end of this thesis an index as well as a comprehensive bibliography. The ¯rst two chapters cover the notions that are basic for this work. Chapter 1 ¯xes the main notation and gives a survey of important results from toric geometry. Chapter 2 gives an exposition of toric Fano varieties and classes of singu- larities that appear naturally when trying to desingularize toric Fano varieties. Here we set up the dictionary of convex-geometric and algebraic-geometric no- tions: Fano polytopes correspond to toric Fano varieties, smooth Fano polytopes to nonsingular toric Fano varieties and canonical (respectively terminal) Fano polytopes to toric Fano varieties with canonical (respectively terminal) singu- larities. Moreover simplicial Fano polytopes are associated to Q-factorial toric Fano varieties. Chapter 3 is the heart of this thesis. Here the main objects of study are in- troduced: Reexive polytopes corresponding to Gorenstein toric Fano varieties. At the beginning two elementary technical tools are investigated and gener- alized that have already been used to successfully investigate and classify non- singular toric Fano varieties [Bat99, Sat00, Deb03, Cas03b]. The ¯rst one, that is especially useful in lower dimensions, is the projection map. We prove some general facts about projections of reexive polytopes (Prop. 3.2.2), thereby we can relate the properties of Gorenstein toric Fano varieties to that of lower- dimensional toric Fano varieties. As an application we present generalizations to mild singularities of an algebraic-geometric result due to Batyrev [Bat99] stating that the anticanonical class of a torus-invariant prime divisor of a non- singular toric Fano variety is always numerically e®ective (Cor. 3.2.7, Prop. 3.2.9). Moreover we get as a trivial corollary (Lemma 3.5.6) that a lattice point on the boundary that has lattice distance one from a facet F has to be con- tained in a facet F 0 intersecting F in a codimension two face. Previously this observation could be proven by Debarre in [Deb03] only in the case of a smooth Fano polytope by using a lattice basis among the vertices of F . The second important tool is the notion of primitive collections and rela- tions. It was introduced by Batyrev in [Bat91] to completely describe smooth Fano polytopes and has been essential for his classi¯cation of nonsingular toric Fano 4-folds [Bat99]. In general this tool is not applicable for reexive poly- topes, since it uses the existence of lattice bases among the vertices. The special case of a primitive collection of length two corresponds to a pair of lattice points on the boundary that do not lie in a common facet. This situation is extremely important and was investigated by Casagrande in [Cas03a] to prove some strong restrictions on smooth Fano polytopes. Now the author has been able to suit- ably generalize this notion to reexive polytopes (Prop. 3.3.1) and apply it successfully to show that the same restrictions also hold for simplicial reexive polytopes. In algebraic-geometric language the corresponding statement reads Introduction 11 as follows: the Picard number of a Q-factorial Gorenstein toric Fano variety exceeds the Picard number of a torus-invariant prime divisor at most by three (Cor.
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