Cotangent cohomology of rational surface singularities (Q1805886)

Let \((Y,y)\) be a germ of a complex algebraic surface having a rational singularity, \(A={\mathcal O}_{Y,y}\). In this paper the authors investigate the André-Quillen (or cotangent cohomology) modules \(T^n(A/ \mathbb{C},A)\) (which will be simply denoted by \(T^n_Y)\). These are their main results: If \((Y,y)\) and \((Y',y')\) are germs as above (i.e., corresponding to rational surface singularities), having the same multiplicity \(d\), then \(\mu(T^n_Y) =\mu(T^n_{Y'})\) for \(n\geq 2\), where \(\mu(M)= \dim_\mathbb{C} N/ {\mathcal N} N\) denotes the minimum number of generators of a module \(N\) over as local ring \((R,{\mathcal N},\mathbb{C})\). Moreover, if the projectivized tangent cone of \((Y, y)\) has only hypersurface singularities then, for \(n\geq 3\), the \(A\)-module \(T^n_Y\) is annihilated by the maximal ideal \({\mathcal M}\) and \(\mu(T^n_Y)= \dim_\mathbb{C} T^n_Y/ {\mathcal M}T^n_Y\). This will be the case, for instance, when \((Y, y)\) is a quotient singularity. Moreover, in this latter case, the authors obtain a rather explicit description of the Poincaré series \(\sum(\dim T^n_Y) t^n\). The authors use a mixture of various techniques. They compare the objects under consideration to the analogous ones for general hyperplane sections (which are related to the so-called partition curves \(H(d_1, \dots, d_r))\), and (by further sections) to those of ``fat points''. On the other hand, by deformation arguments, they may apply explicit results valid for the vertex of the cone over a rational normal curve, also discussed in the paper. In their calculation, they use the fact that (under suitable assumptions) the cotangent cohomology can be computed from the Harrison cohomology \(\text{Harr}^\bullet(A/ \mathbb{C},A)\), which is technically simpler (this is reviewed in the article). They also use a general projection of \(Y\) on a plane (which they call ``Noether normalization''), which allows them to view \(A\) as a finite module over a regular local ring. More explicit results about Poincaré series are obtained for cones over normal rational curves and for the fat point \(\text{Spec}(\mathbb{C}\oplus V)\), where \(V\) is a finite dimensional complex vector space (and the multiplication on \(V\) is trivial).