Loci in Quotients by Finite Groups , Pointwise Stabilizers andthe

Let KV ] G be the invariant ring of a nite 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 decrease if one passes from KV ] G to KV ] G 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 diierence between depth and dimension , and the type. From this, theorems of Steinberg, Serre, Nakajima, Kac and Watanabe, Smith, and the author follow, which say that if KV ] G is a polynomial ring, a hypersurface, a complete intersection, or Cohen-Macaulay, then the same is true for KV ] G U. Furthermore, KV ] G U inherits the Gorenstein property from KV ] G. We give an algorithm which transforms generators of KV ] G into generators of KV ] G U. Let P be one of the properties mentioned above. We consider the locus of P in V= =G := Spec ? KV ] G and prove that for x 2 Spec (KV ]) with image x 0 in V= =G, the local ring KV ] G x 0 has the property P if and only if P holds for the invariant ring KV ] Gx 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 KV ] G is Buchsbaum if and only if it is Cohen-Macaulay. This proves a conjecture of Campbell et al..