Singularity content

We show that a cyclic quotient surface singularity S can be decomposed, in a precise sense, into a number of elementary T-singularities together with a cyclic quotient surface singularity called the residue of S. A normal surface X with isolated cyclic quotient singularities {S_i} admits a Q-Gorenstein partial smoothing to a surface with singularities given by the residues of the S_i. We define the singularity content of a Fano lattice polygon P: this records the total number of elementary T-singularities and the residues of the corresponding toric Fano surface X_P. We express the degree of X_P in terms of the singularity content of P; give a formula for the Hilbert series of X_P in terms of singularity content; and show that singularity content is an invariant of P under mutation.