New rational cubic fourfolds arising from Cremona transformations

TL;DR

This paper demonstrates that for very general cubic fourfolds of discriminant 20, Cremona transformations can produce birational maps, revealing new rational examples and advancing understanding of their equivalence.

Contribution

It introduces a method using Cremona transformations to establish birational equivalences and find new rational cubic fourfolds.

Findings

Cremona transformations produce birational maps for certain cubic fourfolds.

New rational cubic fourfolds are identified.

Affirmative answer to Fourier-Mukai and birational equivalence for specific cases.

Abstract

Are Fourier-Mukai equivalent cubic fourfolds birationally equivalent? We obtain an affirmative answer to this question for very general cubic fourfolds of discriminant 20, where we produce birational maps via the Cremona transformation defined by the Veronese surface. By studying how these maps act on the cubics known to be rational, we surprisingly found new rational examples.