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. Moreover, by studying how these
maps act on the cubics known to be rational, we found new rational examples.Comment: 42 pages. Comments are welcome