A Characterization of Some Fano 4-folds Through Conic Fibrations

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 $\delta _{X}=3$.

Fiberwise bimeromorphic maps of conic bundles

Given a holomorphic conic bundle without sections, we show that the orders of finite groups acting by its fiberwise bimeromorphic transformations are bounded. This provides an analog of a similar result obtained by Bandman and Zarhin for quasi-projective conic bundles.

Stable rationality of higher dimensional conic bundles

We prove that a very general nonsingular conic bundle $X\rightarrow\mathbb{P}^{n-1}$ embedded in a projective vector bundle of rank $3$ over $\mathbb{P}^{n-1}$ is not stably rational if the anti-canonical divisor of $X$ is not ample and $n\geq 3$. Comment: Final version. To appear in Epijournal de Geometrie Algebrique