Flipping surfaces

Abstract: We study semistable extremal threefold neighborhoods following earlier work of Mori, Koll\'ar, and Prokhorov. We classify possible flips and extend Mori's algorithm for computing flips of extremal neighborhoods of type k2A to more general neighborhoods of type k1A. In fact we show that they belong to the same deformation family as k2A, and we explicitly construct the universal family of extremal neighborhoods. This construction follows very closely Mori's division algorithm, which we interpret as a sequence of mutations in the cluster algebra of rank 2 with general coefficients. We identify, in the versal deformation space of a cyclic quotient singularity, the locus of deformations such that the total space admits a (terminal) antiflip. We show that these deformations come from at most two irreducible components of the versal deformation space. As an application, we give an algorithm for computing stable one-parameter degenerations of smooth projective surfaces (under some conditions) and describe several components of the Koll\'ar-Shepherd-Barron boundary of the moduli space of smooth canonically polarized surfaces of geometric genus zero.