Arithmetic properties of the first secant variety to a projective variety

TL;DR

This paper investigates the algebraic and geometric properties of the first secant variety of a projective variety, establishing conditions for projective normality, regularity, and Cohen-Macaulayness, and relating secant variety properties to those of the original variety.

Contribution

It provides explicit positivity conditions ensuring secant varieties are projectively normal, Cohen-Macaulay, and satisfy certain syzygy properties, linking the properties of the variety to its secant.

Findings

Secant variety is projectively normal under positivity conditions.

Conditions for secant variety to be arithmetically Cohen-Macaulay are established.

Secant variety satisfies property N_{3,p} if the original variety satisfies N_{p+2dim(X)}.

Abstract

Under an explicit positivity condition, we show the first secant variety of a linearly normal smooth variety is projectively normal, give results on the regularity of the ideal of the secant variety, and give conditions on the variety that are equivalent to the secant variety being arithmetically Cohen-Macaulay. Under this same condition, we then show that if $X$ satisfies $N_{p+2\dim(X)}$, then the secant variety satisfies $N_{3,p}$.