A remark on Fano 4-folds having (3,1)-type extremal contractions

Abstract

Let X be the blow-up of a smooth projective 4-fold Y along a smooth curve C and let E be the exceptional divisor. Assume that X is a Fano manifold and has an elementary extremal contraction $\varphi :X\to Z$ of (3,1)-type such that E is $\varphi$-ample (recall that a contraction map for a 4-fold is called (3,1)-type if the exceptional locus is a divisor and its image is a curve). We show that if the exceptional divisor of $\varphi$ is smooth, then Y is isomorphic to ${\mathbb{P}}^4$ and C is an elliptic curve of degree 4.

8 pages

Cited by in corpus (1)

• On Fano manifolds with a birational contraction sending a divisor to a curve