Birationally rigid singular double quadrics and double cubics

TL;DR

This paper demonstrates that a broad class of Fano double quadrics and cubics are factorial and birationally superrigid, implying they are non-rational, using advanced methods of maximal singularities and codimension estimates.

Contribution

It establishes the birational superrigidity and factoriality of a large class of Fano double quadrics and cubics, expanding the application of the method of maximal singularities.

Findings

Proves factoriality and birational superrigidity for many Fano double quadrics and cubics.

Shows these varieties admit no non-trivial rationally connected fibrations.

Provides estimates on the codimension of the set of such varieties.

Abstract

In this paper a large class of Fano double quadrics and cubics are shown to be factorial and birationally superrigid, in particular they admit no non-trivial structure of a fibration with rationally connected fibres and are therefore non-rational. This is shown using the "Method of maximal singularities" of Iskovskikh and Manin, expanded upon by Pukhlikov. In addition, an estimate of the codimension of the set of such varieties is calculated.