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.
"Fano 4-folds with rational fibrations." Algebra Number Theory 14 (3) 787 - 813, 2020. https://doi.org/10.2140/ant.2020.14.787