# Results about moduli of surfaces

There are early successes of the moduli theory - the construction and compactification of the moduli spaces of curves $\overline{\mathcal{M}}_g$ .

I want to study about the moduli of algebraic surfaces. What are known results about the moduli of surfaces? What kind of surface does have known moduli space? Do you give any reference?

• This is a vast topic. For surfaces of general type, see the introduction of the upcoming book of Kollar on the subject (web.math.princeton.edu/~kollar/book/chap1.pdf) for a clear view of the additional difficulties that arise. Dec 19, 2014 at 7:58

As already said by Simon in his comment, this is a very vast topic.

Let us stick for simplicity to the case of smooth surfaces $S$ of general type: in this case it is well known that $h^0(S, \, T_S)=0$, hence $\textrm{Aut}(S)$ is a finite group and the moduli functor $\textrm{Def}_S$ is prorepresentable.

The existence of a quasiprojective coarse moduli space $\mathfrak{M}$ for such surfaces modulo birational equivalence was proven by Gieseker in [G77] using GIT theory and the birationality of $5$-canonical map proven by Bombieri in [B73]. It follows that for fixed values of $\chi(\mathcal{O}_S)$ and $K_S^2$ the space $\mathfrak{M}_{\chi, \, K^2}$ has a finite number of irreducible components.

As in the case of $\mathcal{M}_g$, locally in a neighborhood of a point $[S]$ the moduli space $\mathfrak{M}_{\chi, \, K^2}$ is given by a quotient of the base $\textrm{Def}(S)$ of the Kuranishi family of $S$ by the finite group $\textrm{Aut}(S)$. However, in contrast with the case of curves, often patologies arise, for instence $\textrm{Def}(S)$ can be non reduced. This was first explained by Catanese ([C89]) in the case when $K_S$ is not ample, whereas Vakil later proved that such phenomena can be regarded as a particular case of a more general situation that he called the Murphy's law for moduli spaces ([V06]).

In recent years, many people studied possible compactifications of $\mathfrak{M}_{\chi, \, K^2}$. For instance, Kollar and Shepherd-Barron proposed in [KSB88] to add surfaces with semi-log canonical singularities and ample $K_S$, that they called stable surfaces by analogy with stable curves. Using this construction, Alexeev and Pardini were for instance able to provide explicit compactifications of the moduli space of Burniat and Campedelli surfaces ([AP09]).

Here is a (short) bibliography on the subject:

[B73] E. Bombieri: Canonical models for surfaces of general type, Publications Mathématiques de l'IHES 42, Issue 1 (1973), 171-219.

[G77] D. Gieseker: Global moduli for surfaces of general tipe, Invent. math 43 (1977), 233-282.

[C84] F. Catanese: On the moduli spaces of surfaces of general type, J. Differential geometry 19 (1984), 483-515.

[C89] F. Catanese: Everywhere non reduced moduli spaces, Invent. math. 98 (1989), 293-310.

[V06] R. Vakil: Murphy's Law in algebraic geometry: Badly-behaved deformation spaces, Invent. Math. 164 (2006), 569-590.

[KSB88] J. Kollar and N. I- Shepherd-Barron: Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299-338.

[AP09] V. Alexeev, R. Pardini: Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces, arXiv:0901.4431.

• Nice answer! In Catanese's second article: riduced => reduced. (I can't do 1-letter edits…)
– jmc
Dec 19, 2014 at 9:58