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Seshadri Constants and Fujita's Conjecture Via Positive Seshadri Constants and Fujita's Conjecture via Positive Characteristic Methods by Takumi Murayama A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in the University of Michigan 2019 Doctoral Committee: Professor Mircea Mustat¸˘a,Chair Professor Ratindranath Akhoury Professor Melvin Hochster Professor Mattias Jonsson Professor Karen E. Smith ρ = radius 1 given norm on k r− T = p(E(r)) r x0 { } ρ / k∗ Type 3 ∈ | | a Type 2 Type 4 | | x y ∨ [y, x0] y Type 2 x 0 Type 1 points 0 b a D(r) cΓ u ·| u v Spv(A) restriction ⊆ t abstract extension Corr. to a valuation ring Spv(Aa) R Frac(G[a 1]/q) s − 1 ⊂ 1 Spv(G[a ]) dominating (G[a− ]/q)p/q − supp Spec(Aa) dominant supp q ··· a Spec(G[a 1]) e a 1 − − p q Uα X Uβ p n n analytic Vβ C 2 C Vα ⊆ Pk ⊇ Takumi Murayama [email protected] ORCID iD: 0000-0002-8404-8540 c Takumi Murayama 2019 To my family ii Acknowledgments First of all, I am tremendously grateful to my advisor Mircea Mustat¸˘afor his constant support throughout the last five years. I learned much of the philosophy and many of the methods behind the results in this thesis from him, and I am very thankful for his patience and guidance as I worked out the many moving pieces that came together to become this thesis. I would also like to thank the rest of my doctoral committee. I have benefited greatly from taking courses from Ratindranath Akhoury, Melvin Hochster, Mattias Jonsson, and Karen E. Smith, and I am especially glad that they are willing to discuss physics or mathematics with me when I have questions about their work or the classes they are teaching. My intellectual debt to Mel and Karen is particularly apparent in this thesis, since much of the underlying commutative-algebraic theory was developed by Mel and his collaborators, and this theory was first applied to algebraic geometry by Karen among others. I would also like to thank Bhargav Bhatt, from whom I also learned a lot of mathematics through courses and in the seminars that he often led. Next, I would like to thank my collaborators Rankeya Datta, Yajnaseni Dutta, Mihai Fulger, Jeffrey C. Lagarias, Lance E. Miller, David Harry Richman, and Jakub Witaszek for their willingness to work on mathematics with me. While only some of our joint work appears explicitly in this thesis, the ideas I learned through conversations with them have had a substantial impact on the work presented here. I would also like to extend my gratitude toward my fellow colleagues at Michigan for many useful conversations, including (but not limited to) Harold Blum, Eric Canton, Francesca Gandini, Jack Jeffries, Zhan Jiang, Hyung Kyu Jun, Devlin Mallory, Eamon Quinlan-Gallego, Ashwath Rabindranath, Emanuel Reinecke, Matthew Stevenson, Robert M. Walker, Rachel Webb, and Ming Zhang. I am particularly grateful to Farrah Yhee, without whose support I would have had a much harder and definitely more stressful time as a graduate student. I would also like to thank Javier Carvajal-Rojas, Alessandro iii De Stefani, Lawrence Ein, Krishna Hanumanthu, Mitsuyasu Hashimoto, J´anosKoll´ar, Alex K¨uronya, Yuchen Liu, Linquan Ma, Zsolt Patakfalvi, Thomas Polstra, Mihnea Popa, Kenta Sato, Karl Schwede, Junchao Shentu, Daniel Smolkin, Shunsuke Takagi, Hiromu Tanaka, Kevin Tucker, and Ziquan Zhuang for helpful discussions about material in this thesis through the years. Finally, I would like to thank my parents, sisters, grandparents, and other relatives for their constant support throughout my life. They have always been supportive of me working in mathematics, and of my career and life choices. I hope that my grandparents in particular are proud of me, even though all of them probably have no idea what I work on, and some of them did not have the chance to witness me receiving my Ph.D. This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-1265256 and DMS-1501461. iv Table of Contents Dedication ii Acknowledgments iii List of Figures viii List of Tables ix List of Appendices x List of Symbols xi Abstract xiv Chapter 1. Introduction 1 1.1. Outline . .9 1.2. Notation and conventions . 10 Chapter 2. Motivation and examples 11 2.1. Fujita's conjecture . 11 2.2. Seshadri constants . 16 2.3. A relative Fujita-type conjecture . 23 2.4. Difficulties in positive characteristic . 24 2.4.1. Proof of Theorem B . 25 2.4.2. Raynaud's counterexample to Kodaira vanishing . 30 Chapter 3. Characterizations of projective space 36 3.1. Background . 37 3.2. Proof of Theorem A . 41 Chapter 4. Preliminaries in arbitrary characteristic 46 4.1. Morphisms essentially of finite type . 46 4.2. Cartier and Weil divisors . 49 4.3. Reflexive sheaves . 51 v 4.4. Dualizing complexes and Grothendieck duality . 52 4.5. Base ideals and base loci . 53 4.6. Asymptotic invariants of line bundles . 55 4.6.1. Stable base loci . 55 4.6.2. Augmented base loci . 56 4.6.3. Asymptotic cohomological functions . 59 4.6.4. Restricted volumes . 61 4.7. Log pairs and log triples . 62 4.8. Singularities of pairs and triples . 62 4.8.1. Log canonical thresholds . 64 4.9. Multiplier ideals . 66 Chapter 5. Preliminaries in positive characteristic 70 5.1. Conventions on the Frobenius morphism . 70 5.2. The pigeonhole principle . 71 5.3. F -finite schemes . 73 5.4. F -singularities of pairs and triples . 74 5.4.1. The trace of Frobenius . 77 5.4.2. F -pure thresholds . 80 5.5. Test ideals . 81 5.6. Reduction modulo p ............................ 89 5.6.1. Singularities vs. F -singularities . 93 Chapter 6. The ampleness criterion of de Fernex{K¨uronya{Lazarsfeld 96 6.1. Motivation and statement . 97 6.2. A lemma on base loci . 99 6.3. Proof of Theorem E . 105 Chapter 7. Moving Seshadri constants 111 7.1. Definition and basic properties . 111 7.2. Alternative descriptions . 113 7.2.1. Nakamaye's description . 114 7.2.2. A description in terms of jet separation . 116 7.3. A generalization of Theorem B . 127 7.3.1. Proof in positive characteristic . 128 7.3.2. Proof in characteristic zero . 133 Chapter 8. The Angehrn{Siu theorem 138 8.1. The lifting theorem . 139 8.2. Constructing singular divisors and proof of Theorem D . 145 vi Appendices 155 Bibliography 173 vii List of Figures 1.1. Raphael's School of Athens (1509{1511) . .2 1.2. Kobe Port Tower in the Kobe harbor (2006) . .3 2.1. Koll´ar'sexample (Example 2.1.6) . 13 n 2.2. Computing the Seshadri constant of the hyperplane class on Pk ...... 17 2.3. Miranda's example (Example 2.2.9) . 21 2.4. Raynaud's example (Example 2.4.4) . 32 n 3.1. Mori's characterization of Pk ......................... 38 4.1. Log resolution of a cuspidal cubic . 66 5.1. Singularities vs. F -singularities . 94 6.1. Asymptotic cohomological functions on an abelian surface . 98 6.2. Hypercohomology spectral sequence computing Hj(X; (mL rA)) . 109 OX − A.1. Relationships between different classes of F -singularities . 161 viii List of Tables 2.1. Known cases of Fujita's freeness conjecture over the complex numbers . 16 5.1. Some properties preserved under spreading out . 91 5.2. Some properties preserved under reduction modulo p ............ 93 A.1. Proofs of relationships between different classes of F -singularities . 162 ix List of Appendices Appendix A. F-singularities for non-F-finite rings 155 Appendix B. The gamma construction of Hochster{Huneke 163 B.1. Construction and main result . 163 B.2. Applications . 170 B.2.1. Openness of F -singularities . 170 B.2.2. F -singularities for rings essentially of finite type . 171 x List of Symbols Symbols are grouped into three groups, depending on whether they start with punctuation, Greek letters, or Latin letters. Symbol Description ( )! exceptional pullback of Grothendieck duality, 52 − ( )(∗ ) tight closure, 88, 155 − −e ( )[p ] eth Frobenius power of an ideal or module, 71, 88 − ( )◦ complement of the union of minimal primes in a ring, 62 − ( )_ dual of a sheaf, 51 − k-numerical equivalence ≡k k k-linear equivalence (∼Ldim V V ) intersection product, 10 · (X; ∆; aλ) log triple, 62 D • round-up of a divisor, 50 dDe round-down of a divisor, 50 bD c complete linear system, 54 jV j linear system, 54 "j (Dj ; x) Seshadri constant, 17 "( D ; x) moving Seshadri constant, 111 " k( kD ; x) jet separation description for "( D ; x), 117 jet k k k k "N ( D ; x) Nakamaye's description for "( D ; x), 114 ` k k k k "F (D; x) `th Frobenius{Seshadri constant, 132 ∆ "F -sig(D; x) F -signature Seshadri constant, 144 τ(X; ∆; at) test ideal of a triple, 82, 84 τ(X; ∆; aλ) asymptotic test ideal of a triple, 87 τ(X; ∆; t • D ) test ideal of a Cartier divisor, 87 τ(X; ∆; λ· j Dj ) asymptotic test ideal of a Q-Cartier divisor, 88 · k k ΩX cotangent bundle !X canonical bundle or sheaf, 53 !X• (normalized) dualizing complex, 53 a(E; X; ∆; at) discrepancy of E with respect to a triple, 63 xi Symbol Description a graded family of ideals, 54 • a (D) graded family of ideals associated to a divisor, 54 • AnnR M annihilator of a module b( V ) base ideal of a linear system, 54 B(jDj) stable base locus of a divisor, 55 B+(D) augmented base locus of a divisor, 56 x BigRf g(X) subcone of big cone consisting of ξ such that x = B+(ξ), 112 Bs( V ) base scheme of a linear system, 54 2 j j Bs( V )red base locus of a linear system, 54 C j j complex numbers Cartk(X) group of k-Cartier divisors, 49 deg f degree of generically finite map, 147 deg L degree of a line bundle or divisor on a curve Dqc(X) derived category of quasi-coherent sheaves, 52 + Dqc(X) bounded-below derived category of quasi-coherent sheaves, 52 e(R) Hilbert{Samuel multiplicity
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