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 Σk of a curve of g≥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.