Vol. 14, No. 3, 2020

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Fano 4-folds with rational fibrations

Cinzia Casagrande

Vol. 14 (2020), No. 3, 787–813

We study (smooth, complex) Fano 4-folds X having a rational contraction of fiber type, that is, a rational map X −−→ Y that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to the existence of a nonzero, nonbig movable divisor on X. Our main result is that if Y is not 1 or 2, then the Picard number ρX of X is at most 18, with equality only if X is a product of surfaces. We also show that if a Fano 4-fold X has a dominant rational map X −−→ Z, regular and proper on an open subset of X, with dim(Z) = 3, then either X is a product of surfaces, or ρX is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry.

Fano 4-folds, Mori dream spaces, birational geometry, MMP
Mathematical Subject Classification 2010
Primary: 14J45
Secondary: 14E30, 14J35
Received: 27 February 2019
Revised: 17 September 2019
Accepted: 8 November 2019
Published: 1 June 2020
Cinzia Casagrande
Dipartimento di Matematica
Università di Torino