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July 2018 Looijenga’s conjecture via integral-affine geometry
Philip Engel
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J. Differential Geom. 109(3): 467-495 (July 2018). DOI: 10.4310/jdg/1531188193

Abstract

A cusp singularity is a surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In 1981, Looijenga proved that whenever a cusp singularity is smoothable, the minimal resolution of the dual cusp is an anticanonical divisor of some smooth rational surface. He conjectured the converse. Recent work of Gross, Hacking, and Keel has proven Looijenga’s conjecture using methods from mirror symmetry. This paper provides an alternative proof of Looijenga’s conjecture based on a combinatorial criterion for smoothability given by Friedman and Miranda in 1983.

Funding Statement

Research partially supported by NSF grant DMS-1502585.

Citation

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Philip Engel. "Looijenga’s conjecture via integral-affine geometry." J. Differential Geom. 109 (3) 467 - 495, July 2018. https://doi.org/10.4310/jdg/1531188193

Information

Received: 27 January 2016; Published: July 2018
First available in Project Euclid: 10 July 2018

zbMATH: 06917314
MathSciNet: MR3825608
Digital Object Identifier: 10.4310/jdg/1531188193

Rights: Copyright © 2018 Lehigh University

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Vol.109 • No. 3 • July 2018
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