Unirationality of Ueno-type manifold $X_{4,6}$ (following F. Catanese, K. Oguiso, A. Verra)

An Ueno-type manifold is a minimal resolution of singularities of the quotient of the manifold $E(6)^n$ by cyclic group of order $6$, where $E(6)$ is the elliptic curve given by the equation $y^2z = x^3 - z^3$ in $P^2$ and the group acts on each factor as $(x : y : z) \mapsto (\omega x : -y : z)$ ($\omega = e^{\frac{2\pi}{3}}$). We prove that Ueno-type manifold $X_{4,6}$ is unirational if the ground field $k$ is not of characteristic $3$ and contains $\omega$. By some technical computation we show that unirationality of $X_{4,6}$ follows from unirationality of certain cubic surface over the functional field $k(s_3, s_4)$. Then we prove unirationality of this cubic surface.