Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: An algebro-geometric approach
-
Enrico Arbarello
,
Giulio Codogni
and
Giuseppe Pareschi
Abstract
We give completely algebro-geometric proofs of a theorem by T. Shiota, and of a theorem by I. Krichever, characterizing Jacobians of algebraic curves among all irreducible principally polarized abelian varieties. Shiota’s characterization is given in terms of the KP equation. Krichever’s characterization is given in terms of trisecant lines to the Kummer variety. Here we treat the case of flexes and degenerate trisecants. The basic tool we use is a theorem we prove asserting that the base locus of the linear system associated to an effective line bundle on an abelian variety is reduced. This result allows us to remove all the extra assumptions that were introduced in the theorems by the first author, C. De Concini, G.Marini, and O. Debarre, in order to achieve algebro-geometric proofs of the results above.
Funding statement: The last two named authors acknowledge the MIUR “Excellence Department Project”, awarded to the Department of Mathematics, University of Rome, Tor Vergata, CUP E83C18000100006, and the PRIN 2017 “Advances in Moduli Theory and Birational Classifictation”.
Acknowledgements
We thank Robert Auffarth, Olivier Debarre, Thomas Krämer, Gianni Marini, Giulia Saccà, and Riccardo Salvati Manni, for interesting exchanges on the subject of this paper. We also thank the referee for some useful suggestions.
References
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Articles in the same Issue
- Frontmatter
- The nondegenerate generalized Kähler Calabi–Yau problem
- Parity sheaves and Smith theory
- Modular symbols for Teichmüller curves
- Eisenstein series and the top degree cohomology of arithmetic subgroups of SLn/ℚ
- Higher-page Bott–Chern and Aeppli cohomologies and applications
- Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation
- Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: An algebro-geometric approach
- Polynomially convex embeddings of odd-dimensional closed manifolds
Articles in the same Issue
- Frontmatter
- The nondegenerate generalized Kähler Calabi–Yau problem
- Parity sheaves and Smith theory
- Modular symbols for Teichmüller curves
- Eisenstein series and the top degree cohomology of arithmetic subgroups of SLn/ℚ
- Higher-page Bott–Chern and Aeppli cohomologies and applications
- Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation
- Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: An algebro-geometric approach
- Polynomially convex embeddings of odd-dimensional closed manifolds
