Rationality and the FML invariant (Q2871007)

The Makar-Limanov invariant (defined for a variety \(X/k\) to be \(\mathrm{ML}(X):= \bigcap_H k[X]^H\) where \(H\) runs over all the morphisms \(\mathbb{G}_{\mathrm{a}}\rightarrow\mathrm{Aut}(X)\)) is very useful to distinguish varieties from the affine space. Unfortunately, regarding the problem of the rationality of a given variety, this invariant does not contain the right information. In [J. Algebra 324, No. 12, 3653--3665 (2010; Zbl 1219.14069)], \textit{A. Liendo} introduced a version of the ML invariant which seems to be better suited with the problem of rationality. This refined version is defined as: NEWLINE\[NEWLINE\mathrm{FML}(X) := \bigcap_H\mathrm{Frac}\left(k[X]^H\right).NEWLINE\]NEWLINE In [loc. cit.], Liendo conjectured that the equality \(\mathrm{FML}(X) = k\) should be equivalent to the rationality of \(X\) (at least in characteristic zero) ans he proves this conjecture in some special cases. The aim of this paper is to disprove this conjecture in general. A family of counterexamples is constructed using properties of the so-called \textit{special} linear groups (introduced by \textit{J. P. Serre} in [Séminaires Chevalley, exposé 1 (1958)]). It also heavily relies on the existence of finite groups exhibited in [\textit{D. J. Saltman}, Invent. Math. 77, 71--84 (1984; Zbl 0546.14014)] in relation with the Noether problem.