Syzygies of secant varieties of curves of genus 2

Abstract

Ein, Niu and Park showed in [ENP20] that if the degree of the line bundle $L$ on a curve of genus $g$ is at least $2g+2k+1$, the $k$-th secant variety of the curve via the embedding defined by the complete linear system of $L$ is normal, projectively normal and arithmetically Cohen-Macaulay, and they also proved some vanishing of the Betti diagrams. However, the length of the linear strand of weight $k+1$ of the resolution of the secant variety ${\Sigma }_k$ of a curve of $g\ge 2$ is still mysterious. In this paper we calculate the complete Betti diagrams of the secant varieties of curves of genus $2$ using Boij-Söderberg theory. The main idea is to find the pure diagrams that contribute to the Betti diagram of the secant variety via calculating some special positions of the Betti diagram.

14 pages