Vol. 16, No. 6, 2022

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Explicit computation of symmetric differentials and its application to quasihyperbolicity

Nils Bruin, Jordan Thomas and Anthony Várilly-Alvarado

Vol. 16 (2022), No. 6, 1377–1405
Abstract

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 A1-singularities, and use our work to show that Barth’s decic, Sarti’s surface, and the surface parametrizing 3 × 3 magic squares of squares are all algebraically quasihyperbolic.

Keywords
algebraic hyperbolicity, nodal surfaces, symmetric differentials
Mathematical Subject Classification
Primary: 14J60, 14Q10
Secondary: 14J25, 14J29, 14M10
Supplementary material

Scripts for computations supporting Theorems 1.1 and 1.2

Milestones
Received: 14 April 2020
Revised: 9 April 2021
Accepted: 5 October 2021
Published: 27 September 2022
Authors
Nils Bruin
Department of Mathematics
Simon Fraser University
Burnaby, BC
Canada
Jordan Thomas
Department of Mathematics
Courant Institute of Mathematical Sciences
New York University
New York, NY
Anthony Várilly-Alvarado
Department of Mathematics
Rice University
Houston, TX
United States