A Characterization of Some Fano 4-folds Through Conic Fibrations

@article{Montero2018ACO,
  title={A Characterization of Some Fano 4-folds Through Conic Fibrations},
  author={Pedro Montero and Eleonora A. Romano},
  journal={International Mathematics Research Notices},
  year={2018}
}
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… 

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