Nonrational varieties with unirational parametrizations of coprime degrees

We show that there exists a $2$-dimensional family of smooth cubic threefolds admitting unirational parametrizations of coprime degrees. This together with Clemens–Griffiths’ work solves the long standing open problem whether there exists a nonrational variety with unirational parametrizations of coprime degrees. Our proof uses a new approach, called the Noether–Cremona method, for determining the rationality of quotients of hypersurfaces.