# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 136, 2016, pp. 111–136**

**DOI: 10.4171/RSMUP/136-9**

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

Giancarlo Urzúa^{[1]}(1) Pontificia Universidad Católica de Chile, Santiago de Chile, Chile

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.

*Keywords: *Surfaces of general type, moduli spaces, minimal model program, $\mathbb Q$-Gorenstein smoothings

Urzúa Giancarlo: $\mathbb Q$-Gorenstein smoothings of surfaces and degenerations of curves. *Rend. Sem. Mat. Univ. Padova* 136 (2016), 111-136. doi: 10.4171/RSMUP/136-9