Invariant Theory for Unipotent Groups and an Algorithm for Computing Invariants

Abstract

Let X=Spec B be an affine variety over a field of arbitrary characteristic, and suppose that there exists an action of a unipotent group (possibly neither smooth nor connected). The fundamental results are as follows. (1) An algorithm for computing invariants is given, by means of introducing a degree in the ring of functions of the variety, relative to the action. Therefore an algorithmic construction of the quotient, in a certain open set, is obtained. In the case of a Galois extension, k ↪ B=K, which is cyclic of degree p=char k (that is, such that the unipotent group is G=Z/pZ), an element of minimal degree becomes an Artin–Schreier radical, and the method for computing invariants gives, in particular, the expression for any element of K in terms of these radicals, with an explicit formula. This replaces the well-known formula of Lagrange (which is valid only when the degree of the extension and the characteristic are relatively prime) in the case of an extension of degree p=char k. (2) In this paper we give an effective construction of a stable open subset where there is a quotient. In this sense we obtain an algebraic local criterion for the existence of a quotient in a neighbourhood. It is proved (provided the variety is normal) that, in the following cases, such an open set is the greatest one that admits a quotient: when the action is such that the orbits have dimension less than or equal to 1 (arbitrary characteristic) and, in particular, for any action of the additive group Ga; in characteristic 0, when the action is proper (obtained from the results of Fauntleroy) or the group is abelian. 1991 Mathematics Subject Classification: primary 14L30; secondary 14D25, 14D20.