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 $A_1$-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 quasihyperbolic.

我们开发了显式技术，通过对称微分施加的约束来研究奇异曲面的代数准双曲性。我们应用这些方法来证明 Barth 六性曲面上的有理曲线，除了一些众所周知的曲线外，必须至少经过四个奇点，并且第 1 类曲线必须经过至少两个奇点。在对完美长方体进行表面分类时，我们的方法表明，除了一些众所周知的曲线之外，有理曲线必须至少通过七个奇点，并且第 1 类曲线必须通过至少两个。

我们还改进了曲面上对称微分空间维度的下界${\mathrm{一个}}_1$-奇点，并使用我们的工作来证明 Barth 的 decic、Sarti 的曲面和曲面参数化$3\times 3$平方的幻方都是代数拟双曲线的。