Families of curves and variation in moduli

Abstract

In this paper we study the class of smooth complex projective varieties B such that any modular morphism B → M_g  is constant for any g ≥ 2, giving structural properties and examples. Then we investigate the concept of the moduli dimension of a variety B; we bound it by the dimension of the maximal rationally connected quotient of B. In the end we consider also (generically smooth) families of curves of compact type over rational and elliptic curves.