Gregor Kemper
Universitaet Heidelberg

"Loci in Quotients by Finite Groups, Pointwise Stabilizers and the
Buchsbaum Property",
Preprint 2000-10, IWR, Heidelberg, 2000.

Abstract:

  Let K[V]^G be the invariant ring of a finite linear group G <=
  GL(V), and let G_U 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 K[V]^G to K[V]^{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 difference 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 K[V]^G is a polynomial ring, a hypersurface, a
  complete intersection, or Cohen-Macaulay, then the same is true for
  K[V]^{G_U}.  Furthermore, K[V]^{G_U} inherits the Gorenstein
  property from K[V]^G. We give an algorithm which transforms
  generators of K[V]^G into generators of K[V]^{G_U}.
  <P>
  Let P be one of the properties mentioned above. We consider the
  locus of P in V // G := Spec(K[V]^G) and prove that for x in
  Spec(K[V]) with image x' in V // G, the local ring K[V]^G_{x'} has
  the property P if and only if P holds for the invariant ring
  K[V]^{G_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]^G is Buchsbaum if and only if it is
  Cohen-Macaulay. This proves a conjecture of Campbell at el..