Rationality and the Fml Invariant

We construct counterexamples to the rationality conjecture regarding the new version of the Makar-Limanov invariant introduced in [Li2]. Let k be an algebraically closed field. Below variety means algebraic variety over k in the sense of Serre (so algebraic group means algebraic group over k). We use standard notation and conventions of [Bo] and [Sp]. In particular, given a variety X, we denote by k[X] and k(X) respectively the algebra of regular functions and the field of rational functions on X. Action of algebraic group on variety means algebraic action. Recall that the Makar-Limanov invariant of a variety X is the subalgebra ML(X) := ⋂ H k[X] H (1) of k[X], whereH in (1) runs over the images of all algebraic homomorphisms Ga → Aut(X), see [Fr, Chap. 9]. The usefulness of the ML invariant in applications to geometric problems has been amply demonstrated over the last two decades. The highlight is its role in proving that the Koras–Russell threefold is not isomorphic to C that, in turn, is crucial in proving the Linearization Problem for C∗-actions on C. The ML invariant serves for distinguishing some affine varieties from the affine space A whose ML invariant is trivial, i.e., ML(A) = k. However, there are nonrational affine varieties with trivial ML invariant: such singular varieties are constructed in [Li1, Sect. 4.2] and smooth in [Po, Example 1.22]. By [Li2, Thm. 4.2], if char k = 0 and X is an irreducible affine variety of dimension > 2, then