Deformations of quasi-homogeneous surface singularities (Q1106900)

Let \(A=\oplus A_ i\) be a finitely generated \({\mathbb{C}}\)-algebra. So it has a natural filtration by ideals \(I_ j=\oplus_{i\geq j}A_ i.\) So one may define a deformation of \((A\bullet,I\bullet)\) over an artin local \({\mathbb{C}}\)-algebra as a filtered S-algebra \((\bar A,\bar I\bullet)\) with an isomorphism \(\bar A\otimes_ S{\mathbb{C}}\to A\), inducing isomorphisms \(\bar I_ n\otimes {\mathbb{C}}\to I_ n\) for all n. This with the appropriate equivalence defines a functor Def \(+\), deformations of non- negative weight. Further, assume that A is a normal domain with an isolated singularity. If we let \(B=\oplus I_ mt^ m\subset A[t]\), then B is an A-graded algebra and \(X=Proj(B)\) has a natural structural morphism to Spec(A). Let \(Y=Proj(A)\). Then there is a natural embedding of Y in X. One may now define a functor \(ES_ X\), equisingular deformations of X, along Y. In this paper, the author proves that there is a natural isomorphism of functors \(Def^+\to ES_ X\). Various corollaries about rational singularities and cones over smooth projective varieties are deduced.