Graph invariants and the positivity of the height of the Gross-Schoen cycle for some curves (Q2655181)

Let \(X\) be a smooth projective curve over a function field \(K\) of a smooth projective curve \(B\) over a field \(k\), and suppose that there exists a semistable model \(f:\mathbb{X\rightarrow}B\) of \(X\). The main purpose of this article is to compute the height \(<\Delta,\Delta>\) of the modified diagonal cycle \(\Delta\) on \(X^{3}\) defined in [\textit{B. H. Gross, C. Schoen}, Ann. Inst. Fourier 45, No. 3, 649--679 (1995; Zbl 0822.14015)], when the genus \(g\) of \(X\) is equal to three. A crucial ingredient for his computation is a formula \(<\Delta,\Delta>=(2g+1)/(2g-2)(\omega_{\mathbb{X}/B}\cdot\omega_{\mathbb{X}/B})-\sum_{y\in B(k)}\psi(\mathbb{X}_{y})\), proved in [\textit{S. Zhang}, Invent. Math. 179, No. 1, 1--73 (2010; Zbl 1193.14031)], where \(\psi(\mathbb{X}_{y})\) depends only on the reduction graph with the associated polarization at \(y\). As an application he gives a sufficient condition for \(<\Delta,\Delta> \) being positive.