Nonrational varieties with unirational parametrizations of coprime degrees

TL;DR

This paper constructs a 2-dimensional family of smooth cubic threefolds with unirational parametrizations of coprime degrees, solving a long-standing open problem about nonrational varieties with such properties.

Contribution

It introduces the Noether–Cremona method to determine the rationality of quotients of hypersurfaces, demonstrating the existence of nonrational varieties with coprime unirational degrees.

Findings

Existence of a 2D family of cubic threefolds with coprime unirational degrees

Solution to the open problem on nonrational varieties with coprime unirational parametrizations

Development of the Noether–Cremona method for analyzing rationality

Abstract

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.