G_a actions on Cⁿ

Abstract

Rational actions of the additive group of complex numbers on complex n space are considered. A ring theoretic criterion for properness is given, along with ideal theoretic criteria for local triviality of such actions. The relationship between local triviality and flatness of the polynomial ring over its subring of G, invariants is investigated.

A proper 𝐺ₐ action on 𝐶⁵ which is not locally trivial

“…It was proved in [ 1 ] that o is proper if and only if ö is surjective, and locally trivial if and only if im(<5) n C[X]Ga generates the unit ideal in C[X]. In the latter case the action admits a quasiaffine geometric quotient.…”

“…An easy consequence of the surjectivity of ô is the absence of fixed points. Moreover, for proper Ga actions we have shown that the action is locally trivial if and only if the ring extension QX]01" «-» C[X] is flat, and equivariantly trivial if and only if the extension is faithfully flat [1]. If there is a slice for the action, i.e.…”

“…A proper action of Ga onI = C" is locally trivial if C [X] is a flat extension of its subring of Ga invariants and equivariantly trivial if the extension is faithfully flat [1]. Moreover, under those conditions the geometric exists as a quasiaffine variety.…”

Triangular 𝐺ₐ actions on 𝐂⁴

“…It was recently shown [2] that C 0 is finitely generated for any triangular action on C 4 . In the special case under consideration, we show that Y ∼ = Spec C 0 .…”

Free $G_a$ actions on $C^3$

“…In the special case where X = C n and the dimension of U is at least n − 2, a famous result of Miyanishi [10,Theorem 4] gives the requisite condition on the ring of invariants. For n = 3, U = G a , and the action proper, an Euler characteristic argument shows that the action is not merely locally trivial, but in fact conjugate to a translation [4]. Fauntleroy and Magid prove in [6] that a free G a action on any factorial quasiaffine surface is conjugate to a translation.…”