Negative curves on surfaces and applications to hermitian locally symmetric spaces

Let $S$ be a smooth projective surface over a field $k$. In this paper we show that if $\mathcal{F}$ is a set of irreducible curves on $S$ such that sufficiently many curves in $\mathcal{F}$ have negative self-intersection, then there exist two curves $C_1, C_2 \in \mathcal{F}$ and integers $a, b\ge 1$ such that the effective divisor $a C_1 + b C_2$ has positive self-intersection. When $k = \mathbb{C}$, this implies that the image of the map $\pi_1(C_1 \cup C_2) \to \pi_1(S)$ of topological fundamental groups has finite index and that the Albanese variety of $S$ is a quotient of the direct sum of the Jacobians of the normalizations of $C_1$ and $C_2$. We then give some applications of these results to the geometry of smooth projective surfaces uniformized by either the product of two upper half planes or the complex hyperbolic plane.