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H 1 Special rational fibrations in Fano 4-folds

Auteurs : Casagrande, Cinzia (Auteur de la Conférence)
CIRM (Editeur )

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Résumé : Smooth, complex Fano 4-folds are not classified, and we still lack a good understanding of their general properties. We focus on Fano 4-folds with large second Betti number $b_{2}$, studied via birational geometry and the detailed analysis of their contractions and rational contractions (we recall that a contraction is a morphism with connected fibers onto a normal projective variety, and a rational contraction is given by a sequence of flips followed by a contraction). The main result that we want to present is the following: let $X$ be a Fano 4-fold having a nonconstant rational contraction $X --> Y$ of fiber type. Then either $b_{2}(X)$ is at most 18, with equality only for a product of surfaces, or $Y$ is $\mathbb{P}^{1}$ or $\mathbb{P}^{2}$. The proof is achieved by reducing to the case of "special" rational contractions of fiber type. We will explain this notion and give an idea of the techniques that are used.

Keywords : Fano varieties, 4-folds, birational geometry

Codes MSC :
14E30 - Minimal model program (Mori theory, extremal rays)
14J35 - Algebraic $4$-folds
14J45 - Fano varieties

 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 21/10/2019 Date de captation : 03/10/2019 Collection : Research talks Format : MP4 Durée : 01:00:50 Domaine : Algebraic & Complex Geometry Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2019-10-03_Casagrande.mp4 Informations sur la rencontre Nom de la rencontre : The Geometry of Algebraic Varieties / Géométrie des variétés algébriquesOrganisateurs de la rencontre : Benoist, Olivier ; Jiang, Zhi ; Voisin, ClaireDates : 30/09/2019 - 04/10/2019 Année de la rencontre : 2019 URL Congrès : https://conferences.cirm-math.fr/2069.htmlCitation Data DOI : 10.24350/CIRM.V.19565303 Cite this video as: Casagrande, Cinzia (2019). Special rational fibrations in Fano 4-folds. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19565303 URI : http://dx.doi.org/10.24350/CIRM.V.19565303

### Voir aussi

Bibliographie

1. CASAGRANDE, Cinzia. Fano 4-folds with rational fibrations. arXiv preprint arXiv:1902.01835, 2019. - https://arxiv.org/abs/1902.01835

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