On the rationality of the moduli space of Lüroth quartics (Q443952)

A \textit{Lüroth quartic} is a quartic plane curve containing a complete pentagon, i.e. a set of \(10\) points which are the pairwise intersections of \(5\) lines. The locus of Lüroth quartics has a closure \(L\), in the projective space \(\mathbb P^{14}\) of all quartics, which is an irreducible hypersurface of degree \(54\). The quotient of \(L\) modulo the action of SL\({}_3\) is the moduli space \(\mathcal L\) of Lüroth quartics. The authors prove that \(\mathcal L\) is rational. Indeed, they prove that \(\mathcal L\) is birational to the quotient mod SL\({}_3\) of the space of \textit{Clebsch quartics}, i.e. quartics that can be written as a sum of \(5\) powers of linear forms. The rationality of the latter space is proven by using methods of representation theory. The authors also prove that a covering space of \(\mathcal L\), which parametrizes Bateman configurations of seven-tuples of points, is rational.