Fano 4-folds with rational fibrations

Authors :: Cinzia Casagrande
Source :: Algebra Number Theory 14, no. 3 (2020), 787-813
Publication Year :: 2020
Publisher :: Mathematical Sciences Publishers, 2020.
Abstract: 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 non-zero, non-big movable divisor on X. Our main result is that if Y is not P^1 or P^2, then the Picard number rho(X) 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 rho(X) is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry.<br />25 pages. Minor changes. To appear in Algebra & Number Theory