Deformations of cusp singularities
-- Robert Friedman, April 1, 2016

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Cusp singularities are certain normal surface singularities with a
rich geometry and deformation theory. In particular their smoothing
components are very closely connected to the moduli of certain
rational surfaces. In 1981, Looijenga gave a necessary condition for a
cusp singularity to be smoothable and conjectured that this condition
was also sufficient, a conjecture recently proved by
Gross-Hacking-Keel and Engel. This talk describes ongoing joint work
with Engel. We define an invariant &lambda; for every semi-stable
K-trivial model of a one parameter smoothing of a cusp singularity and
show that all possible values of the invariant arise. Using this
result, we characterize those rational double point singularities
which are adjacent to cusp singularities.