A Curve of Secants to the Kummer Variety from Degenerate Points

TL;DR

This paper establishes a geometric condition linking secant planes to the Kummer variety with the existence of a curve of secants, using theta functions and inductive reasoning.

Contribution

It introduces a new method to construct a curve of secants to the Kummer variety from degenerate points via theta function equations.

Findings

Existence of a curve of secants under specific geometric conditions.

Construction of equations in terms of theta functions from curve germs.

Inductive approach to extend initial secant conditions to entire curves.

Abstract

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.