Quotients by Connected Solvable Groups

Abstract

This paper introduces the notion of an excellent quotient, which is stronger than a universal geometric quotient. The main result is that for an action of a connected solvable group $G$ on an affine scheme Spec$(R)$ there exists a semi-invariant $f$ such that Spec$(R_f) \to$ Spec$((R_f)^G)$ is an excellent quotient. The paper contains an algorithm for computing $f$ and $(R_f)^G$. If $R$ is a polynomial ring over a field, the algorithm requires no Gr\"obner basis computations, and it also computes a presentation of $(R_f)^G$. In this case, $(R_f)^G$ is a complete intersection. The existence of an excellent quotient extends to actions on quasi-affine schemes.