Seshadri Constants and Fujita's Conjecture via Positive Characteristic Methods
arXiv:1905.03839
Abstract
In 1988, Fujita conjectured that there is an effective and uniform way to turn an ample line bundle on a smooth projective variety into a globally generated or very ample line bundle. We study Fujita's conjecture using Seshadri constants, which were first introduced by Demailly in 1992 with the hope that they could be used to prove cases of Fujita's conjecture. While examples of Miranda seemed to indicate that Seshadri constants could not be used to prove Fujita's conjecture, we present a new approach to Fujita's conjecture using Seshadri constants and positive characteristic methods. Our technique recovers some known results toward Fujita's conjecture over the complex numbers, without the use of vanishing theorems, and proves new results for complex varieties with singularities. Instead of vanishing theorems, we use positive characteristic techniques related to the Frobenius-Seshadri constants introduced by Mustaţă-Schwede and the author. As an application of our results, we give a characterization of projective space using Seshadri constants in positive characteristic, which was proved in characteristic zero by Bauer and Szemberg.
arXiv comment206 pages, 12 figures, 4 tables. PhD thesis, University of Michigan, May 2019. Includes material from arXiv:1701.00511 and arXiv:1809.01217
References in corpus (21)
- Invariance of Plurigenera
- F-signature exists
- Seshadri constants on algebraic surfaces
- F-singularities of pairs and Inversion of Adjunction of arbitrary codimension
- A survey of test ideals
- Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems
- The non-nef locus in positive characteristic
- Valuations and Frobenius
- Effective Matsusaka's Theorem for surfaces in characteristic p
- Differentiability of non-archimedean volumes and non-archimedean Monge-Ampère equations (with an appendix by Robert Lazarsfeld)
- Effective freeness of adjoint line bundles
- Effective generation and twisted weak positivity of direct images
- Frobenius-Seshadri constants and characterizations of projective space
- The gamma construction and asymptotic invariants of line bundles over arbitrary fields
- Fano varieties with large Seshadri constants
- On the behavior of singularities at the -pure threshold
- The Angehrn-Siu Type Effective Freeness For Quasi-Log Canonical Pairs
- Bertini Theorems for -signature and Hilbert-Kunz multiplicity
- Cohomological characterizations of the complex projective space
- On the Simultaneously Generation of Jets of the Adjoint Bundles
- On the Picard numbers of abelian varieties in positive characteristic