We develop explicit techniques to investigate algebraic quasihyperbolicity 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
-singularities,
and use our work to show that Barth’s decic, Sarti’s surface, and the surface parametrizing
magic
squares of squares are all algebraically quasihyperbolic.
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