Singularities and syzygies of secant varieties of nonsingular projective curves

TL;DR

This paper thoroughly studies secant varieties of nonsingular projective curves, proving their singularities, Cohen–Macaulay property, and syzygy conditions under certain degree bounds, and introduces a new vanishing theorem for Cartesian products.

Contribution

It settles several conjectures on secant varieties, classifies their singularities, and establishes a vanishing theorem, advancing understanding of their algebraic and geometric properties.

Findings

Secant varieties have normal Du Bois singularities under certain degree conditions.

Secant varieties are arithmetically Cohen–Macaulay and satisfy property N_{k+2, p}.

A new vanishing theorem for Cartesian products of curves is proved.

Abstract

In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus $g$ is greater than $2g+2k+p$ for nonnegative integers $k$ and $p$, then the $k$-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen--Macaulay, and satisfies the property $N_{k+2, p}$. In addition, the singularities of the secant varieties are further classified according to the genus of the curve, and the Castelnuovo--Mumford regularities are also obtained as well. As one of the main technical ingredients, we establish a vanishing theorem on the Cartesian products of the curve, which may have independent interests and may find applications elsewhere.