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

TL;DR

This paper investigates Fano 4-folds with specific extremal contractions, proving that under certain smoothness conditions, the original manifold is projective 4-space and the blown-up curve is elliptic of degree 4.

Contribution

It establishes a classification result for Fano 4-folds with (3,1)-type extremal contractions under smoothness assumptions, linking the structure to projective space and elliptic curves.

Findings

Y is isomorphic to P^4

C is an elliptic curve of degree 4

X exhibits a specific geometric structure under the assumptions

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 $\phi: X \to Z$ of (3,1)-type such that E is $\phi$-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 $\phi$ is smooth, then Y is isomorphic to $\mathbb{P}^{4}$ and C is an elliptic curve of degree 4.