A characterization of some Fano 4-folds through conic fibrations

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 $f:X\to Y$ 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 ${\delta }_X=3$.

21 pages. v2: major revision