Loci in Quotients by Finite Groups, Pointwise Stabilizers and the Buchsbaum Property

Let K[V ] be the invariant ring of a finite linear group G ≤ GL(V ), and let GU be the pointwise stabilizer of a subspace U ≤ V . We prove that the following numbers associated to the invariant ring do not increase if one passes from K[V ] to K[V ]U : the minimal number of homogeneous generators, the maximal degree of the generators, the number of syzygies and other Betti numbers, the complete intersection defect, the difference between depth and dimension, and the type. From this, theorems of Steinberg, Serre, Nakajima, Kac and Watanabe, and the author follow, which say that if K[V ] is a polynomial ring, a hypersurface, a complete intersection, or Cohen-Macaulay, then the same is true for K[V ]U . Furthermore, K[V ]U inherits the Gorenstein property from K[V ]. We give an algorithm which transforms generators of K[V ] into generators of K[V ]U . Let P be one of the properties mentioned above. We consider the locus of P in V//G := Spec ( K[V ] ) and prove that for x ∈ Spec (K[V ]) with image x′ in V//G, the local ring K[V ]Gx′ has the property P if and only if P holds for the invariant ring K[V ]x of the point stabilizer. Using this, we prove that the non-Cohen-Macaulay locus in V//G is either empty, or it has dimension at least one and codimension at least 3. From this we deduce that K[V ] is Buchsbaum if and only if it is Cohen-Macaulay. This proves a conjecture of Campbell et al..