Secants to the Kummer Variety and the minimal Cohomological Class

Abstract

We prove that, under certain conditions, the existence of a curve of $(m+2)$-secants to the Kummer variety of an indecomposable principally polarized abelian variety $X$, represents $m$-times the minimal cohomological class in $X$. In the case of $m=2$, we find an involution of such curve which proves that \(X\) is a Prym variety by results of Krichever-Grushevsky. This continues the work of Beauville and Debarre, who asked about the relation between some geometric properties of abelian varieties in a way to obtain a stratification of its moduli space.