Equations and syzygies of the first secant variety to a smooth curve

TL;DR

This paper investigates the algebraic properties of the first secant variety to a smooth curve, establishing conditions under which it satisfies certain syzygy properties related to its embedding.

Contribution

It proves that for a linearly normal smooth curve with sufficiently high degree, the secant variety satisfies the property N_{3,p}, advancing understanding of its algebraic structure.

Findings

Secant variety satisfies N_{3,p} under specified degree conditions

Provides new criteria linking curve degree and secant variety syzygies

Enhances knowledge of algebraic relations in secant varieties

Abstract

We show that if C is a linearly normal smooth curve embedded by a line bundle of degree at least 2g+3+p then the secant variety to the curve satisfies N_{3,p}.