$\mathbb Q$-Gorenstein smoothings of surfaces and degenerations of curves

Abstract

In this paper we mainly describe \mathbb Q -Gorenstein smoothings of projective surfaces with only Wahl singularities which have birational fibers. For instance, these degenerations appear in normal degenerations of \mathbb P^2 , and in boundary divisors of the KSBA compactification of the moduli space of surfaces of general type [15]. We give an explicit description of them as smooth deformations plus 3-fold birational operations, through the flips and divisorial contractions in [9]. We interpret the continuous part (smooth deformations) as degenerations of certain curves in the general fiber. At the end, we work out examples happening in the KSBA boundary for invariants K^2=1 , p_g=0 , and \pi_1=0 using plane curves.