arXiv subjects

José Alejandro Aburto

Publications and source records attributed to José Alejandro Aburto.

2 recordsLinked to original sources

arXiv2026

A Curve of Secants to the Kummer Variety from Degenerate Points

We prove that, under certain geometric conditions, that only $m-1$ different non-degenerate $(m+2)$-secant $m$-planes plus one degenerate $(m+2)$-secant $m$-plane to the Kummer variety implies the existence of a curve of ${(m+2)}$-secants to the Kummer variety. This is done by constructing a set of equations in terms of theta functions from the germ of a curve on the described points. The relation between those equations allows to proceed by induction to get the entire desired curve since the first of them is equivalent to the hypothesis that we ask.

math.AG↗

arXiv2026

Secants to the Kummer Variety and the minimal Cohomological Class

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.

math.AG↗