Can one obtain surfaces with interesting invariants as resolutions of singular surfaces?

(Perhaps a not very well defined question)

Let $(S_t)_t$ be a (flat) family of compact complex surfaces. Assume the generic member is smooth while $S_0$ has isolated singularities. As the simplest case consider a degenerating family of surfaces in $\Bbb P^3$. Let $\tilde{S}\to S_0$ be the minimal resolution of singularities.

1. Can one obtain in this way some interesting surfaces (in the sense of geography problem)? (References on works/summaries?)

2. Suppose I have some inequality among the local singularity invariants of $S_0$. The inequality will translate into some inequality among the invariants of $\tilde{S}$. Are there examples when such singularity inequalities are useful?

The answer to your first question is definitely yes. In fact, many interesting examples of smooth, complex algebraic surfaces are given by desingularization of singular ones. The subject is too broad to be fully treated in a MO post. However, let me just recall the paper by Stephan Endraß, Ulf Persson and Jan Stevens

"Surfaces with triple points", http://arxiv.org/abs/math/0010163

in which the authors consider the effect of imposing a finite number of ordinary triple points to the invariants of a surface in $\mathbb{P}^3$. In fact, in contrast to the case of ordinary double points (classified by Du Val as those which "do not effect the conditions of adjunction"), when we impose one or more ordinary triple points both the invariants of the surface and its type in the classification may change.

The purpose of the paper mentioned above is to point out the intricate geometry of examples with many triple points, and how it fits with the general classification of surfaces.

• Thanks! I'll read the paper, meanwhile a more specific question. Suppose I have an upper bound on the "defect of the geometric genus" (i.e. the drop down of $p_g$ in the family) in terms of the change of $c_2$ in the family. (Or in singularity theory terms, a bound on the genus of the singularity in terms of its Milnor number.) Can it be helpful in producing some new surfaces? Or it can only serve as an obstruction to the method? Commented Apr 23, 2011 at 3:02
• In my experience, this kind of bounds only serve as an obstruction to the method. In order to produce some new surfaces, one usually has to find some clever construction. The $\mathbb{Q}$-Gorenstein smoothing method of Lee and Park mentioned in EOP's answer is a good example. Commented Apr 23, 2011 at 8:56
• Dear Francesco, thanks for the answer. One related question: when is the irregularity of the surface preserved in a (flat) degenerating family? e.g. from the examples of surfaces in $\Bbb P^3$ with at most four singularities I had an impression that $q(S)$ might be preserved. But in the paper of E.P.S. they consider quintics in $\Bbb P^3$. A quintic with 5 ordinary triple points has irregularity 1. :( Any criteria for $q(S)$ to be preserved? Commented Apr 24, 2011 at 16:17

Recently many interesting complex surfaces of general type with $p_g=q=0$ and small (topological, algberaic) fundamental groups are constructed via the so-called $\mathbb{Q}$-Gorenstein smoothing method. You may refer the paper [Invent. Math. 2007] by Yongnam Lee and Jongil Park.

• The $\mathbb{Q}$-Gorenstein method is in some sense "dual" to the one proposed by the OP. In fact, in the OP's construction the general fiber of the family is "uninteresting", whereas the central fibre is an "interesting" surface. In Lee-Park's method, instead, the central fibre is "uninteresting" (usually, it is a rational surface with carefully chosen quotient singularities) whereas the general fibre (the "smoothing") is the "interesting" one Commented Apr 23, 2011 at 9:12