Cardinality of special locus of a non-isotrivial family of algebraic curves over positive characteristic

Abstract: In this talk, I will report a recent joint work with Jun Lu and Kang Zuo on the cardinality of the p-rank zero locus of a non-isotrivial semi-stable family of algebraic curves of genus greater than one. We shall give an upper bound of such a locus in terms of the fiber genus, base genus , number of singular fibers and the characterisic p, when the family does not lie entirely in the p-rank zero stratum of the moduli space of algebraic curves over algebraically closed field of characteristic p.