Subvarieties of moduli spaces

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In this paper we try to decide along algebraic lines whether moduli spaces of abelian varieties or of algebraic curves contain complete subvarieties. In Theorem (1.1) we consider abelian varieties and curves of genus three in characteristic p :~ 0. In Section 2 we use monodromy in order to show certain families of abelian varieties up to a purely inseparable isogeny are isotrivial; probably the results (2.1) and (2.2) are special cases of more general facts about / -adic monodromy. In Section 4 we answer a question raised by Martin concerning a possible generalization of the fact that two supersingular elliptic curves over an algebraically closed field (of characteristic p) are isogenous. We abbreviate abelian variety(ies) by AV; we use X t for the dual, and 2 for the formal group of an AV X.