A classical and very natural problem in algebraic geometry is to determine, given a projective variety Y, which varieties admit a (non-trivial) rational map to Y.

After giving some examples and motivations, we will restrict our attention to the locus of curves of fixed genus mapping to Y. We will state two theorems giving an upper bound of the dimension of this locus when Y is a very general hypersurface in $\mathbb{P}^{n+1}$ of high enough degree and when Y is an abelian variety. We will prove the theorem regarding hypersurfaces, giving particular attention to a statement we need in the argument, which we feel is of independent interest. This is based on a joint work with J. Lam and F. Moretti.