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 |$f\colon X\to Y$| 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 |$\delta _{X}=3$|⁠.