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Finite generation of Cox rings and relations among Fano 3-folds

Résumé :

Minimal model program provides new insights for the classification of algebraic varieties, and in particular Fanos.
The next step is to study relations among Fano varieties. A particular, and classical, relation of this type is the rationality for a smooth,
or mildly singular, hypersurface $X_d\subset\mathbb{P}^n$ of degree $d$ for $d\leq n$ (still unknown in many cases).

In this talk, using the theory of Sarkisov program (a la Corti), I highlight a strong relation between the finite generation of certain
Cox rings and the existence of birational maps from a given Fano 3-fold. This provides an effective and systematic method of obtaining
birational maps between Mori fibre spaces or proving their non-existence (the birational rigidity in the extreme case). I will report
on joint projects with Anne-Sophie Kaloghiros and Francesco Zucconi for particular cases in both directions.
Thème de recherche :
Algèbre et géométries
Salle :
04