On birational maps from cubic threefolds

We characterise smooth curves in a smooth cubic threefold whose blow-ups produce a weak-Fano threefold. These are curves C of genus g and degree d, such that (i) 2(d −5) ≤ g and d ≤ 6; (ii) C does not admit a 3-secant line in the cubic threefold. Among the list of ten possible such types (g,d), two yield Sarkisov links that are birational selfmaps of the cubic threefold, namely (g,d) = (0,5) and (2,6). Using the link associated with a curve of type (2, 6), we are able to produce the first example of a pseudo-automorphism with dynamical degree greater than 1 on a smooth threefold with Picard number 3. We also prove that the group of birational selfmaps of any smooth cubic threefold contains elements contracting surfaces birational to any given ruled surface.