A characterization of some Fano 4-folds through conic fibrations

TL;DR

This paper characterizes certain Fano 4-folds with Lefschetz defect 3, showing they are either products of del Pezzo surfaces or admit a specific conic bundle structure, and provides explicit examples, especially in the toric case.

Contribution

It introduces a new characterization of Fano 4-folds with Lefschetz defect 3 via conic bundle structures and classifies their possible targets, enriching the understanding of their geometry.

Findings

Fano 4-folds with Lefschetz defect 3 are either products of del Pezzo surfaces or admit a conic bundle structure.

All such varieties are rational.

Explicit examples are provided in the toric case.

Abstract

We find a characterization for Fano 4-folds $X$ with Lefschetz defect $\delta_{X}=3$: besides the product of two del Pezzo surfaces, they correspond to varieties admitting a conic bundle structure $f\colon X\to Y$ with $\rho_{X}-\rho_{Y}=3$. Moreover, we observe that all of these varieties are rational. We give the list of all possible targets of such contractions. Combining our results with the classification of toric Fano $4$-folds due to Batyrev and Sato we provide explicit examples of Fano conic bundles from toric $4$-folds with $\delta_{X}=3$.