May 2021 Smoothings and rational double point adjacencies for cusp singularities
Philip Engel, Robert Friedman
Author Affiliations +
J. Differential Geom. 118(1): 23-100 (May 2021). DOI: 10.4310/jdg/1620272941


A cusp singularity is a surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in dual pairs. Looijenga proved in 1981 that if a cusp singularity is smoothable, the minimal resolution of the dual cusp is the anticanonical divisor of some smooth rational surface. In 1983, the second author and Miranda gave a criterion for smoothability of a cusp singularity, in terms of the existence of a K‑trivial semistable model for the central fiber of such a smoothing. We study these “Type III degenerations” of rational surfaces with an anticanonical divisor—their deformations, birational geometry, and monodromy. Looijenga’s original paper also gave a description of the rational double point configurations to which a cusp singularity deforms, but only in the case where the resolution of the dual cusp has cycle length $5$ or less. We generalize this classification to an arbitrary cusp singularity, giving an explicit construction of a semistable simultaneous resolution of such an adjacency. The main tools of the proof are (1) formulas for the monodromy of a Type III degeneration, (2) a construction via surgeries on integral-affine surfaces of a degeneration with prescribed monodromy, (3) surjectivity of the period map for Type III central fibers, and (4) a theorem of Shepherd–Barron producing a simultaneous contraction to the adjacency.

Funding Statement

P. Engel’s research was partially supported by NSF grant DMS-1502585.


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Philip Engel. Robert Friedman. "Smoothings and rational double point adjacencies for cusp singularities." J. Differential Geom. 118 (1) 23 - 100, May 2021.


Received: 5 November 2016; Published: May 2021
First available in Project Euclid: 7 May 2021

Digital Object Identifier: 10.4310/jdg/1620272941

Rights: Copyright © 2021 Lehigh University


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Vol.118 • No. 1 • May 2021
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