Explicit computation of symmetric differentials and its application to quasi-hyperbolicity

We develop explicit techniques to investigate algebraic quasi-hyperbolicity of singular surfaces through the constraints imposed by symmetric differentials. We apply these methods to prove that rational curves on Barth's sextic surface, apart from some well-known ones, must pass through at least four singularities, and that genus 1 curves must pass through at least two. On the surface classifying perfect cuboids, our methods show that rational curves, again apart from some well-known ones, must pass through at least seven singularities, and that genus 1 curves must pass through at least two. We also improve lower bounds on the dimension of the space of symmetric differentials on surfaces with $A_1$-singularities, and use our work to show that Barth's decic, Sarti's surface, and the surface parametrizing $3\times 3$ magic squares of squares are all algebraically quasi-hyperbolic.