Some recent topics in smoothable stable log surfaces

Abstract

This is an introductory article of the study of smoothable stable log surfaces. First we briefly introduce the projectivity of the moduli functor of smoothable stable log surfaces with given Hilbert function. The positivity of the second Chern class of stable log surfaces and its relation of boundedness are discussed. Also the study of limits of plane curves by using smoothable stable log surfaces, and the construction of surfaces by using smoothable stable surfaces are shortly discussed.