On the birational geometry of Fano 4-folds

Abstract

We study the birational geometry of a Fano fourfold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map $$f: X \dashrightarrow Y$$ , where Y is normal and projective, which factors as a finite sequence of flips, followed by a surjective morphism with connected fibers. Such f is called elementary if ρ X − ρ Y = 1, where ρ is the Picard number. We first give a characterization of non-movable prime divisors in X, when ρ X ≥ 6; this is related to the study of birational and divisorial elementary rational contractions of X. Then we study the rational contractions of fiber type on X which are elementary or, more generally, “quasi-elementary”. The main result is that ρ X ≤ 11 if X has an elementary rational contraction of fiber type, and ρ X ≤ 18 if X has a quasi-elementary rational contraction of fiber type.