Residues and Smoothings of Cyclic Quotient Singularities

This talk will be about cyclic quotient singularities on surfaces. Locally, these are quotients of the affine plane by the action of a cyclic group. The cyclic quotient singularities which admit Q-Gorenstein smoothings are precisely the T-singularities introduced by Kollár--Shepherd-Barron in 1988. At the other extreme are the R-singularities, introduced in joint work with Alexander Kasprzyk. These are Q-Gorenstein rigid. We will discuss how every cyclic quotient singularity can be decomposed into a collection of T-singularities and one possibly empty R-singularity, called the residue. In particular, every cyclic quotient singularity admits a Q-Gorenstein partial smoothing to its residue, extending the observation of Kollár--Shepherd-Barron. Our central tool is the theory of combinatorial mutations, introduced in joint work with Coates--Galkin--Kasprzyk. If time permits, we will discuss how these considerations lead to the notion of singularity content, an invariant of combinatorial mutation which plays an important role in the classification theory of Fano orbifold surfaces.