Syzygies of secant varieties of smooth projective curves and gonality sequences

TL;DR

This paper demonstrates that the gonality sequence of a smooth projective curve can be deduced from the syzygies of its secant varieties, extending the gonality conjecture to higher secant varieties.

Contribution

It establishes a link between gonality sequences and syzygies of secant varieties, generalizing previous results on syzygies of curves.

Findings

Gonality sequence determines the minimal free resolutions of secant varieties.

Main theorem extends gonality conjecture to secant varieties.

Syzygies encode gonality information for large degree embeddings.

Abstract

The purpose of this paper is to prove that one can read off the gonality sequence of a smooth projective curve from syzygies of secant varieties of the curve embedded by a line bundle of sufficiently large degree. More precisely, together with Ein-Niu-Park's theorem, our main result shows that the gonality sequence of a smooth projective curve completely determines the shape of the minimal free resolutions of secant varieties of the curve of sufficiently large degree. This is a natural generalization of the gonality conjecture on syzygies of smooth projective curves established by Ein-Lazarsfeld and Rathmann to the secant varieties.