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

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 λ 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.