On the locus of curves mapping to a fixed target

Suppose $Y$ is a smooth variety equipped with a top form. We prove a simple theorem giving a sharp lower bound on the geometric genus of a family of subvarieties of $Y$, in terms of the dimension of this family. Two elementary applications are presented. On the one hand, we show that for a very general curve $C$ and a very general hypersurface $Y\subset \mathbb P^{n+1}$ of degree $\ge 2n+1$, any map $C \to Y$ is constant. On the other hand, we give a lower bound on the genus of a family of curves with an isotrivial factor in the associated family of Jacobians; we also characterize the families of curves attaining this bound as the families of degree $2$ branched covers of a fixed curve.