The cubo-cubic transformation of the smooth quadric fourfold is very special

The classification of special Cremona transformations is a classical problem that is completely understood when the dimension of the base locus is at most three. This result is the culmination of the work of B. Crauder, S. Katz, L. Ein, N. Shepherd-Barron, and G. Staglianò from 1987 to 2019. However, what do we know about special self-birational transformations of varieties different from projective spaces? A 2021 article by M. Bernardara, E. Fatighenti, L. Manivel, and F. Tanturri, titled "Fano Fourfolds of K3 Type," explores 64 families of Fano fourfolds with a K3-type structure. One of these families, labeled K3-33, gives rise to a special cubo-cubic self-birational transformation of the smooth quadric fourfold. The base locus of this transformation is a non-minimal K3 surface of degree 10 with two skew (-1)-lines, and the base locus of the inverse map is also a non-minimal K3 surface of the same type. However, the two associated K3 surfaces turn out to be non-isomorphic Fourier-Mukai partners. My recent work shows that this is the only special self-birational transformation for a smooth quadric fourfold and explores its geometry. This represents a first step toward the classification of special self-birational transformations of smooth quadrics with a base locus of dimension at most three.