Height and arithmetic intersection for a family of semi-stable curves (Q1585436)

The author proves an arithmetic Hodge index theorem for a family of semi-stable curves (theorem 5.2). The result generalizes Faltings-Hriljac's arithmetic Hodge index theorem for an arithmetic surface [\textit{G. Faltings}, Ann. Math. 119, 387-424 (1984; Zbl 0559.14005); \textit{P. Hriljac}, Am. J. Math. 107, 23-38 (1985; Zbl 0593.14004)]. The main ingredients in the proof are: the arithmetic Riemann-Roch theorem established by \textit{H. Gillet} and \textit{Ch. Soulé} [Invent. Math. 110, 473-543 (1992; Zbl 0777.14008)], and a theory of arithmetic height functions over function fields à la Moriwaki [\textit{A. Moriwaki}, Invent. Math. 140, 101-142 (2000; Zbl 1007.11042)].