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Explicit computation of
symmetric differentials and its application to
quasihyperbolicity
Nils Bruin, Jordan Thomas and Anthony Várilly-Alvarado |
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Vol. 16 (2022), No. 6, 1377–1405
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Abstract |
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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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Keywords
algebraic hyperbolicity, nodal surfaces, symmetric
differentials
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Mathematical Subject Classification
Primary: 14J60, 14Q10
Secondary: 14J25, 14J29, 14M10
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Supplementary material |
Milestones
Received: 14 April 2020
Revised: 9 April 2021
Accepted: 5 October 2021
Published: 27 September 2022
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Authors |
| Nils Bruin | |
| Department of Mathematics Simon Fraser University Burnaby, BC Canada |
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| Jordan Thomas | |
| Department of Mathematics Courant Institute of Mathematical Sciences New York University New York, NY |
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| Anthony Várilly-Alvarado | |
| Department of Mathematics Rice University Houston, TX United States |
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