Weighted Homogeneous Complete Intersections

Abstract

Suppose given a set of weights and degrees defining ℂ^× actions on ℂⁿ and ℂ^p with n ≥ p. Necessary and sufficient conditions are obtained for the existence of an equivariant map f : ℂⁿ → ℂ^p such that f ⁻¹(0) has an isolated singularity at 0. These are somewhat complicated, but simplify if n−p = 0 or 1 or if p = 1. The former case gives conditions for (weighted) homogeneously generated ideals of finite codimension in the ring O _n of germs of holomorphic functions; these are generalised to submodules of finite codimension in free O _n -modules. For maps f as above, there are known formulae for the Poincaré series of the Jacobian algebra and the K-cotangent space; we also have a corresponding formula for the quotient in the submodule case.

For the case of A- (right-left-) equivalence of maps f, the above results can be used to give an algorithm for the Poincaré series of the A-cotangent, space (of a finitely A-determined germ) hi terms of the weights and degrees. The method yields necessary conditions for existence of a finitely A-determined germ which are not, however, sufficient.

To express the condition for K-finite maps, write the source as ℂ^I with weights {w _i | i ∈ I} and the target as ℂ^J with degrees {d _j | j ∈; J}. For A $\subseteq$ I write ℕ(A) for the additive monoid generated by {w _i | i ∈ A} and J (A) = {j ∈ J | d _j ∈ ℕ(A)}; write # to denote cardinality. The condition is that for all A $\subseteq$ I such that #A>#J(A) and all non-empty B $\subseteq$ J\J(A),

$$\# \left\{ {i \in I\backslash A|\exists j \in B{\text{ with }}{d_j} - {w_i} \in \mathbb{N}\left( A \right)} \right\} \geqslant \# A + \# B - \# J\left( A \right) - 1.$$