Fields of G_a Invariants are Ruled

Abstract

The quotient field of the ring of invariants of a rational Ga action on Cn is shown to be ruled. As a consequence, all rational Ga actions on C4 are rationally triangulable. Moreover, if an arbitrary rational Ga action on Cn is doubled to an action of Ga × Ga on C2n, then the doubled action is rationally triangulable.

Generalizations of Samuel's criteria for a ring to be a unique factorization domain

“…We show that condition P(iii) implies that W pb, sppq, tppqq " Ş iě0 ppA `bAq i for every p P P . By condition P(iv), this suffices to prove (6).…”

“…One motivation to consider UFDs of the type given in this section comes from the study of locally nilpotent derivations of polynomial rings C rns for the field C of complex numbers. The kernel A of such a derivation is a UFD of transcendence degree n ´1 over C. It is known that A is quasi-affine [22] and fracpAq is ruled [6]; that A -C rn´1s if 1 ď n ď 3 [18]; and that A is generally non-noetherian if n ě 5 [3,10]. For n " 4, it is further known that A is rational [6] and that there can be no a priori bound on the number of generators needed [4].…”

Ga-actions of fiber type on affine T-varieties

“…The previous proof depends only on the fact that X has at least two non-equivalent LNDs. An alternative proof of the first assertion of Theorem 4.2 can be obtained adapting the argument of Theorem 2.5 in [4].…”

On some properties of elementary derivations in dimension six

“…and it is a well-known fact (see for example Lemma 2:1 of [4]) that ker S −1 D is equal to im where is the homomorphism…”