Equations defining secant varieties: geometry and computation

TL;DR

This paper explores the algebraic and geometric properties of secant varieties of curves, focusing on their defining equations, syzygies, and explicit computations to illustrate underlying geometric principles.

Contribution

It provides an introduction to the algebraic and geometric methods used to study secant varieties, emphasizing explicit examples and elementary cases.

Findings

Connections between curve embeddings and secant variety equations

Explicit computations illustrating geometric principles

Extension of classical results to higher secant varieties

Abstract

In the 1980's, work of Green and Lazarsfeld helped to uncover the beautiful interplay between the geometry of the embedding of a curve and the syzygies of its defining equations. Similar results hold for the first secant variety of a curve, and there is a natural conjectural picture extending to higher secant varieties as well. We present an introduction to the algebra and geometry used in previous work of the authors to study syzygies of secant varieties of curves with an emphasis on examples of explicit computations and elementary cases that illustrate the geometric principles at work.