Rationally integrable vector fields and rational additive group actions (Q2816970)

Given an algebraic variety \(X\) over a field \(k\) of characteristic zero, with field of rational functions \(K_X\), the authors define a rationally integrable derivation on \(X\) to be a \(k\)-derivation \(\partial: K_X \to K_X\) such that the formal exponential map \(f \in K_X \mapsto \sum_n \frac{\partial^n(f)}{n!}t^n \in K_X[[t]]\) factors through the subalgebra \(K_X(t)\) of rational functions. This recovers a notion previously introduced by Makar-Limanov (unpublished lecture notes), but with a formulation that is easier to check in practice. The main result of the paper is that the rationally integrable derivations on \(X\) are in one-to-one correspondence with the rational \(\mathbb G_a\)-actions on \(X\). Then this point of view is used to characterize regular \(\mathbb G_a\)-actions on semi-affine varieties. A few other applications are given, in particular a description of homogeneous rational \(\mathbb G_a\)-actions on toric varieties.