On birational maps from cubic threefolds

Abstract

We characterise smooth curve 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) \(2(d-5) \le g\) and \(d\le 6\); (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.