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A Curve of Secants to the Kummer Variety from Degenerated Points

José Alejandro Aburto

Abstract

We prove that, under certaing geometric conditions, that only \(m-1\) different honest $(m+2)$-secants plus one degenerate $(m+2)$-secant 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.

A Curve of Secants to the Kummer Variety from Degenerated Points

Abstract

We prove that, under certaing geometric conditions, that only different honest -secants plus one degenerate -secant to the Kummer variety implies the existence of a curve of -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.
Paper Structure (6 sections, 3 theorems, 27 equations)

This paper contains 6 sections, 3 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $m$ different $(m+2)$-secants
  4. Preliminaries
  5. Hierarchy
  6. Finite secants implies the existence of a curve

Key Result

Theorem 3.1

The abelian variety $X$ satisfies $\dim_{-a_1-a_2}V_Y>0$ if and only if there exist complex numbers $\alpha_{j,i}$, with $1\leq j\leq m+1$, $j\neq 2$, $i\geq 1$; and constant vector fields $D_1\neq 0, D_2,D_3\ldots$ on $X$ such that the sections $P_s$ defined before, vanish for all positive integers

Theorems & Definitions (7)

  • Definition 2.1
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof