Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: An algebro-geometric approach-Reference-Cited by-同舟云学术

Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: An algebro-geometric approach

Published:2021-05-08 Issue:777 Volume:2021 Page:251-271
ISSN:0075-4102
Container-title:Journal für die reine und angewandte Mathematik (Crelles Journal)
language:en
Short-container-title:

Author:

Arbarello Enrico¹,Codogni Giulio²^ORCID,Pareschi Giuseppe²

Affiliation:

1. Classe di Scienze , Accademia Nazionale dei Lincei , Via della Lungara 10, 00165 Roma , Italy

2. Dipartimento di Matematica , Università Tor Vergata , Via della Ricerca Scientifica 1, 00133 Roma , Italy

Abstract

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.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/crelle-2021-0020/pdf

Reference34 articles.

1. E. Arbarello, Fay’s trisecant formula and a characterization of Jacobian varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick 1985), Proc. Sympos. Pure Math. 46, American Mathematical Society, Providence (1987), 49–61.

2. E. Arbarello and C. De Concini, On a set of equations characterizing Riemann matrices, Ann. of Math. (2) 120 (1984), no. 1, 119–140.

3. E. Arbarello and C. De Concini, An analytical translation of a criterion of Welters and its relation with the K.P. hierarchy, Algebraic geometry, Sitges (Barcelona 1983), Lecture Notes in Math. 1124, Springer, Berlin (1985), 1–20.

4. E. Arbarello and C. De Concini, Another proof of a conjecture of S. P. Novikov on periods of abelian integrals on Riemann surfaces, Duke Math. J. 54 (1987), no. 1, 163–178.

5. E. Arbarello and C. De Concini, Geometrical aspects of the Kadomtsev–Petviashvili equation, Global geometry and mathematical physics (Montecatini Terme 1988), Lecture Notes in Math. 1451, Springer, Berlin (1990), 95–137.