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 $\delta_{X}=3$.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2018
- DOI:
- 10.48550/arXiv.1803.09129
- arXiv:
- arXiv:1803.09129
- Bibcode:
- 2018arXiv180309129M
- Keywords:
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- Mathematics - Algebraic Geometry;
- 14E08;
- 14E30;
- 14J35;
- 14J45
- E-Print:
- 21 pages. v2: major revision