On birational maps from cubic threefolds

Abstract

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) $2(d-5)\le g$ 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 were previously left as open numerical possibilities, namely $(g,d)=(0,5)$ and $(2,6)$. Using the Sarkisov 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.

In the last version, the relation with the work of [CM13] has been developed and the fact that the open subset in the moduli space of curve is dense has been added

References in corpus (2)

• Dynamical degrees of (pseudo)-automorphisms fixing cubic hypersurfaces
• On the existence of certain weak Fano threefolds of Picard number two

Cited by in corpus (1)

• Some properties of dynamical degrees with a view towards cubic fourfolds