Fano 4-folds with a small contraction

TL;DR

This paper establishes an upper bound of 12 on the Picard number for smooth complex Fano 4-folds with small elementary contractions, and describes their geometric structure in the boundary case.

Contribution

It proves a bound on Picard number for Fano 4-folds with small contractions and characterizes their geometry, extending previous classification results.

Findings

Picard number rho(X) is at most 12 for such Fano 4-folds

In the case rho(X)=12, an open subset admits a smooth P^1-fibration

If rho(X)>12, all elementary contractions are divisorial and map to surfaces

Abstract

Let X be a smooth complex Fano 4-fold. We show that if X has a small elementary contraction, then the Picard number rho(X) of X is at most 12. This result is based on a careful study of the geometry of X, on which we give a lot of information. We also show that in the boundary case rho(X)=12 an open subset of X has a smooth fibration with fiber the projective line. Together with previous results, this implies if X is a Fano 4-fold with rho(X)>12, then every elementary contraction of X is divisorial and sends a divisor to a surface. The proof is based on birational geometry and the study of families of rational curves. More precisely the main tools are: the study of families of lines in Fano 4-folds and the construction of divisors covered by lines, a detailed study of fixed prime divisors, the properties of the faces of the effective cone, and a detailed study of rational contractions of fiber type.