Let $X$ be an algebraic curve defined over the complex numbers $\mathbb{C}$ of genus $g > 1$. A theorem of De Franchis states that there exist only finitely many (isomorphism classes of) curves $Y$ defined over $\mathbb{C}$ with genus exceeding one such that there exists a non-constant map $f: X \rightarrow Y$. For a given curve $X$, put $S(X)$ for the set of (isomorphism classes of) curves $Y$ with genus exceeding one such that there exists a non-constant map $f: X \rightarrow Y$.
Now suppose that for two curves $X_1, X_2$ with equal genus $g > 1$, that $S(X_1) = S(X_2)$. What can be said about $X_1, X_2$?