Algebra & Number Theory
- Algebra Number Theory
- Volume 14, Number 3 (2020), 787-813.
Fano 4-folds with rational fibrations
We study (smooth, complex) Fano 4-folds having a rational contraction of fiber type, that is, a rational map 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 . Our main result is that if is not or , then the Picard number of is at most 18, with equality only if is a product of surfaces. We also show that if a Fano 4-fold has a dominant rational map , regular and proper on an open subset of , with , then either is a product of surfaces, or is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry.
Algebra Number Theory, Volume 14, Number 3 (2020), 787-813.
Received: 27 February 2019
Revised: 17 September 2019
Accepted: 8 November 2019
First available in Project Euclid: 2 July 2020
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Primary: 14J45: Fano varieties
Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 14J35: $4$-folds
Casagrande, Cinzia. Fano 4-folds with rational fibrations. Algebra Number Theory 14 (2020), no. 3, 787--813. doi:10.2140/ant.2020.14.787. https://projecteuclid.org/euclid.ant/1593655272