Refined de Jonqui\`eres divisors and secant varieties on algebraic curves

TL;DR

This paper offers a new perspective on secant varieties of algebraic curves by using refined de Jonqui es divisors, leading to both known and novel results on their dimension theory through degeneration techniques.

Contribution

It introduces a reformulation of secant varieties in terms of refined de Jonqui es divisors, providing new insights and results in their dimension analysis.

Findings

Recovered known results on secant varieties.

Obtained new dimension bounds for secant varieties.

Used degeneration arguments to study refined de Jonqui es divisors.

Abstract

The aim of this paper is to provide another perspective on secant varieties on algebraic curves by reformulating the problem in terms of refined de Jonqui\`eres divisors, that is divisors on the curve with prescribed multiplicities and dimensions of their spaces of global sections. We are able to both recover some already known results and to obtain some new statements concerning the dimension theory of secant varieties. We do this via the study of the dimension theory of refined de Jonqui\`eres divisors in some relevant cases and degeneration arguments.