Abstract

(This is part II of a series of two talks.)The moduli space of surfaces of general type has a natural compactification where the boundary points correspond to ``stable surfaces’’. If we restrict attention to surfaces which are branched double covers of certain simple surfaces (like the projective plane or a Hirzebruch surface) then singularities can develop in three ways: the branch curve can degenerate, or the base surface can degenerate, or the branch curve and the base surface can develop singularities at the same point. If the limit of the base surface is toric, one can use toric geometry to understand the stable limit of the double cover, but often degenerations of the plane or a Hirzebruch surface are only almost toric (in a precise sense). Thanks to work of Gross, Hacking and Keel, the same diagrammatic techniques that work for toric degenerations can be applied in this setting, and one can use this to get a full classification of normal stable surfaces for some components of the moduli space. I will explain how this works for octic double planes. This is based on joint work with Angelica Simonetti and Giancarlo Urzua.