In 1987 S. Katz proved that the cubo-cubic transformation of P^3 is the only special Cremona transformation of projective space. It is given by blowing up a smooth curve of degree 6 and genus 3.
A 2021 article by M. Bernardara, E. Fatighenti, L. Manivel and F. Tanturri named "Fano Fourfolds of K3 type" explores 64 families of Fano fourfolds with a K3-type structure. One of them, labeled as K3-33, gives rise to a
cubo-cubic birational transformation of the quadric Q^4 that resembles a lot the example of S. Katz. My recent work shows that this is the only special birational transformation of Q^4 with a surface S as fundamental locus. Moreover, S is a non-minimal K3 surface of degree 10 and the fundamental locus on the target quadric is also a non-minimal K3 surface with the same numerical data as S. Nonetheless, it is possible to show that the K3's are non-isomorphic Fourier-Mukai partners.
