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

TL;DR

This paper aims to characterize Jacobians using the KP equation and geometric properties of Kummer varieties, but its main proof contains a flaw, affecting some key theorems and conclusions.

Contribution

It provides algebro-geometric proofs of characterizations of Jacobians via the KP equation and trisecants, contingent on an unproven theorem.

Findings

Algebro-geometric proofs of Shiota's theorem

Characterizations of Jacobians via trisecants

Dependence on an unproven key theorem

Abstract

This paper is withdrawn since we found a flaw in the proof of Theorem 4, asserting that the base locus of the complete linear system of an ample line bundle on a complex abelian variety is reduced. The error is in page 7, line $ -14$, where we claim that the divisor "mathcal E" on the variety $X$ is linearly equivalent to zero. This is untrue. For instance, it would imply that, for a non-torsion point $x$ on an abelian surface $A$, letting $E_x$, $E_{-x}$, and $E_0$ the exceptional curves in the blow up of $A$ at $x$, $-x$, and $0$, then $2E_0$ is linearly equivalent to $E_x +E_{-x}$, which is easily seen to be false. Therefore Theorem 4 of our paper has to be considered unproven. We still believe that it holds true. All the other arguments of our paper are correct but unfortunately they depend on the above mentioned Theorem 4. To be precise, from Theorem 4 follows Theorem 3, asserting that the scheme $\Sigma(X,\Theta, G)$ of Definition 9 is reduced. The rest of the paper contains algebro-geometric proofs of Shiota's theorem characterizing Jacobians via the KP equation (Section 4), and of Krichever's theorems characterizing Jacobians by the existence of an inflectionary or degenerate trisecant to the Kummer variety embedded in $\mathbb P^{2^g-1}$ (Theorem 18 and Theorem 25). Also these proofs are correct but they depend on a weaker version of Theorem 3, namely on the assertion that the components of codimension two of the scheme $\Sigma(X,\Theta, G)$ are generically reduced. In turn, this weaker version of Theorem 3 would follow, by the same argument used in its proof, from a conjecture by Debarre asserting that the base locus of the complete linear system of an ample line bundle on an abelian variety is generically reduced in codimension two.