A note on $\mathbb{G}_a$-actions in positive characteristic
P M S Sai Krishna
TL;DR
The paper investigates when Miyanishi's zero-characteristic result on the invariants of $\mathbb{G}_a$-actions can be extended to positive characteristic and develops a robust framework around exponential maps to study rigidity of invariants. It provides a concrete sufficient condition—the plinth ideal containing a quasi-basic element—that forces $B^{\delta}=k^{[2]}$ for $B=k^{[3]}$, and proves non-rigidity results for rings of invariants arising from triangular and linear locally nilpotent derivations. By introducing and analyzing commuting exponential maps, the authors establish that the invariants of commuting exponential maps are typically non-rigid and relate these to the weak Abhyankar–Sathaye conjecture; they also prove that, for characteristic zero, the commuting exponential maps conjecture is equivalent to WAS(n), yielding CD(3). Overall, the work links invariant theory of $\mathbb{G}_a$-actions, rigidity phenomena, and classical coordinate problems through a unified exponential-map framework.
Abstract
Miyanishi proved that the ring of invariants of any $\mathbb{G}_a$ action on $\mathbb{A}^3$ is $\mathbb{A}^2$, when the field $k$ has zero characteristic. However, it is not known if this result holds when $k$ has positive characteristic. We provide a sufficient condition under which this result holds in positive characteristic. We also prove the following results related to the rigidity of the ring of invariants of an exponential map of a polynomial ring. (1) Let $B=R^{[n]}$, where $R$ is a $k$-domain and $δ\in \mathrm{EXP}_R(B)$ is a triangular exponential map. Then $B^δ$ is non-rigid. In particular, for any field $k$ of zero characteristic the kernel of any triangular $R$-derivation of $R^{[n]}$ is non-rigid. (2) Let $k$ be a field of zero characteristic and $R$ be a $k$-domain. Then the kernel of any linear locally nilpotent $R$-derivation of $R^{[n]}$ is non-rigid. When $k$ is an algebraically closed of zero characteristic, the commuting derivations conjecture for $k^{[3]}$ has been proved by Maubach and El Kahoui proved that the weak Abhyankar Sathaye conjecture is equivalent to the commuting derivations conjecture. By introducing the notion of commuting exponential maps and formulating the commuting exponential maps conjecture, we show that the weak Abhyankar-Sathaye conjecture is equivalent to the commuting exponential maps conjecture for any field of arbitrary characteristic. In particular, we prove the commuting derivations conjecture$(CD(3))$ for any field of zero characteristic.