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

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 ${langle Delta, Delta rangle = 0}$ .