No abstract

INTRODUCTIONDue to the work in [Brieskorn], [Tjurina], [Artin], [Wahl 2] and [Lipman] one understands very well the notion of simultaneous resolution for rational surface singularities. Given a rational surface singularity X, there exists a smooth parameter space Res representing the functor of deformations of the (minimal) resolution. The family Y ---+ Res contracts to X ---+Res. The fibers yt---+ X 1 are the minimal resolutions of Xt. There is a finite and Galois map Res ---+ Defx (the versa! base space of X). The image is the Artin component, which represents the functor of deformations of X with simultaneous resolution after finite base change, and the covering group W is a reflection group which is also the monodromy group of the component.In the case of rational double points (RDPs) W is the Weyl group of the corresponding root system (Ak, Dk, E6, E1 orEs). In general much information about the deformation, e.g. the discriminant and adjacencies, may be read off the geometry of this covering. The .) The first author conjectured that this was the monodromy cover and we asked ourselves if there was a deformation theoretic explanation. The purpose of this paper is to answer this question.We will show that something similar to the Res ---+ Def picture actually happens for every non-embedded component of the versa! base space of a quotient singularity. (For us a quotient singularity is the singularity of C 2 /G where G C GL(2, C) is a finite subgroup which we can assume to be without pseudo-reflections.) For quotient singularities, the construction of the Artin component is a special case of a procedure involving deformations of certain modifications which we call M-resolutions.The application of threefold theory to deformations of rational singularities as found in [Kollfu--Shepherd-Barron] was important both for discovering M-resolutions and for the proofs of their properties. In fact, our results are anticipated in Theorem 3.5(a) of that paper. Still -the definitions and statement of the main result may be made without reference to that work. The relevant results from [Kollfu--Shepherd-Barron] are postponed to §1.Consider a smoothing of a normal surface singularity X with smooth generic fiber F.The Milnor number of this smoothing is p. = rk H 2 (F) and depends upon the component of the versa! base space of X on which the smoothing appears.1 supported by a "Heisenberg-Stipendium" Be 1078/1-2 of the DFG These singularities are classified (see §1). The simplest example is the cone over the rational normal curve of degree 4. The curve is a hyperplane section of the Veronese surface in P 5 • We get a smoothing of the surface singularity by "sweeping out the cone".Thus the Milnor fiber is P 2 minus a quadric, which has b2 = 0.IT a local deformation of a proper modification Y of a quotient surface singularity X restricts to (p, = 0)-smoothings of the singularities of Y, then the second betti number of the fibers is constant, so there is no monodromy. Hence if the blowing down map to the deformation space of X i...

Simplicial complexes X provide commutative rings A(X) via the StanleyReisner construction. We calculated the cotangent cohomology, i.e., T 1 and T 2 of A(X) in terms of X. These modules provide information about the deformation theory of the algebro geometric objects assigned to X.

We study the deformation theory of projective Stanley-Reisner schemes associated to combinatorial manifolds. We achieve detailed descriptions of first order deformations and obstruction spaces. Versal base spaces are given for certain StanleyReisner surfaces.

We construct degenerations of Mukai varieties and linear sections thereof to special unobstructed Fano Stanley-Reisner schemes corresponding to convex deltahedra. This can be used to find toric degenerations of rank one index one Fano threefolds. In the second part we find many higher dimensional unobstructed Fano and Calabi-Yau Stanley-Reisner schemes. The main result is that the Stanley-Reisner ring of the boundary complex of the dual polytope of the associahedron has trivial T 2 .