The Dynkin diagrams of rational double points

It is well known that the exceptional set in a resolution of a rational surface singularity is a tree of rational curves. We generalize the combinatoric part of this statement to higher dimensions and show that the highest cohomologies of the dual complex associated to a resolution of an isolated rational singularity vanish. We also prove that the dual complex associated to a resolution of an isolated hypersurface singularity is simply connected. As a consequence, we show tha...

The paper is an extended version of the talk which I gave at the XIX Congresso dell'UMI in Bologna in September 2011. The aim of this paper is twofold: first, to give an overview on the recent development in the classification of surfaces of general type with $p_g=q=2$; second, to point out some of the problems that are still open.

Due to a significant error in the main result (pointed out by J. Wahl), the paper has been withdrawn by the authors. A corrected and expanded version is 'Rational blow-downs and smoothings of surface singularities' by A. Stipsicz, Z. Szabo and J. Wahl.

An elementary introduction to the principles of algebraic surgery.

We classify real families of minimal degree rational curves that cover an embedded rational surface. A corollary is that if the projective closure of a smooth surface is not biregular isomorphic to the projective closure of the unit-sphere, then the set of minimal degree rational curves that cover the surface is either empty or of dimension at most two. Moreover, if these curves are of minimal degree over the real numbers, but not over the complex numbers, then almost all the...

In this paper, we classify the configurations of the singular points which appear on the quotients of the projective plane by the $1$-foliations of degree $-1$ in characteristic $2$.

Let p be a singular point of a variety. Consider a resolution where the preimage of p is a simple normal crossing divisor E. The combinatorial structure of E is described by a cell complex D(E), called the dual graph or dual complex of E. It is known that the homotopy type of D(E) depends only on p, not on the resolution chosen. We prove that this homotopy type can be arbitrary. We also describe which homotopy types can be obtained from rational singularities.

Using the structure of the jet schemes of rational double point singularities, we construct "minimal embedded toric resolutions" of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every "minimal embedded toric resolution". We prove that this correspondence is bijective except for the $E_8$ singulartiy. Th...

We study several examples of surfaces with $p_g = q = 2$ and maximal Albanese dimension that are endowed with an irrational fibration.

We study in this work flat surfaces with conical singularities, that is, surfaces provided with a flat structure with conical singular points. Finding good parameters for these surfaces in the general case is an open question. We give an answer to this question in the case of flat structures on pairs of pants with one singular point. The question of decomposability of an arbitrary flat surface into flat pairs of pants is discussed.