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INVARIANT THEORY FOR UNIPOTENT GROUPS AND AN ALGORITHM FOR COMPUTING INVARIANTS

Published online by Cambridge University Press:  19 October 2000

CARLOS SANCHO DE SALAS
Affiliation:
Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1–4, 37008 Salamanca, Spain, sancho@gugu.usal.es
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Abstract

Let $X=\operatorname{Spec} B$ 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, $k\hookrightarrow B=K$, which is cyclic of degree $p=\text{ch} (k)$ (that is, such that the unipotent group is $G={\Bbb Z}/p {\Bbb Z}$), 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 $p=\text{ch}(k)$. (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: \begin{enumerate} \item[(a)] 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 $G_a$; \item[(b)] 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.

Type
Research Article
Copyright
© 1999 London Mathematical Society

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