$\mathbb{Q}$-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 the projective plane, and in boundary divisors of the KSBA compactification of the moduli space of surfaces of general type [KSB88]. We give an explicit description of them as smooth deformations plus 3-fold birational operations, through the flips and divisorial contractions in [HTU13]. 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.