Fano 4-folds with a small contraction

Abstract

Let X be a smooth complex Fano 4-fold. We show that if X has a small elementary contraction, then ρX≤12, where ρX is the Picard number of X. 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 ρX=12 an open subset of X has a smooth fibration with fiber P1. Together with previous results, this implies that if X is a Fano 4-fold with ρX≥13, 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.