The degree of the third secant variety of a smooth curve of genus 2

TL;DR

This paper introduces a new method to compute the degree of the third secant variety of a genus 2 smooth curve in projective space, leveraging scrolls defined by a specific linear system.

Contribution

It presents a novel approach to calculating the degree of the third secant variety using scrolls associated with a g^1_3 linear system on the curve.

Findings

Provides a new computational method for the degree of the third secant variety.

Expresses the secant variety as a union of scrolls defined by a g^1_3.

Applicable for curves with degree d ≥ 8.

Abstract

We give a new method of computation of the degree of the third secant variety of a smooth curve C in P^(d-2) of genus 2 and degree d>=8, using the presentation of the third secant variety as the union of all scrolls that are defined via a g^1_3 on C.