Graph invariants and the positivity of the height of the Gross-Schoen cycle for some curves

Abstract

Let X be a projective curve over a global field K. Gross and Schoen defined a modified diagonal cycle Δ on X ³, and showed that the height ${\langle \Delta, \Delta \rangle}$ is defined in general. Zhang recently proved a formula which describe ${\langle \Delta, \Delta \rangle}$ in terms of the self pairing of the admissible dualizing sheaf and the invariants arising from the reduction graphs. In this note, we calculate explicitly those graph invariants for the reduction graphs of curves of genus 3 and examine the positivity of ${\langle \Delta, \Delta \rangle}$ . We also calculate them for so-called hyperelliptic graphs. As an application, we find a characterization of hyperelliptic curves of genus 3 by the configuration of the reduction graphs and the property ${\langle \Delta, \Delta \rangle = 0}$ .