On birational maps from cubic threefolds

TL;DR

This paper characterizes certain curves in smooth cubic threefolds that lead to weak-Fano threefolds upon blow-up, introduces the first example of a pseudo-automorphism with dynamical degree greater than one, and explores the birational selfmaps of cubic threefolds.

Contribution

It provides a complete characterization of specific curves in cubic threefolds and constructs a novel pseudo-automorphism with high dynamical degree using Sarkisov links.

Findings

Characterized curves leading to weak-Fano threefolds with specific genus and degree.

Constructed the first pseudo-automorphism with dynamical degree > 1 on a smooth threefold with Picard number 3.

Showed that birational selfmaps can contract surfaces birational to any ruled surface.

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.