Fano 4-folds with a small contraction
| Autor: | Casagrande, C. |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let X be a smooth complex Fano 4-fold. We show that if X has a small elementary contraction, then the Picard number rho(X) of X is at most 12. This result is based on a careful study of the geometry of X, on which we give a lot of information. We also show that in the boundary case rho(X)=12 an open subset of X has a smooth fibration with fiber the projective line. Together with previous results, this implies if X is a Fano 4-fold with rho(X)>12, then every elementary contraction of X is divisorial and sends a divisor to a surface. The proof is based on birational geometry and the study of families of rational curves. More precisely the main tools are: the study of families of lines in Fano 4-folds and the construction of divisors covered by lines, a detailed study of fixed prime divisors, the properties of the faces of the effective cone, and a detailed study of rational contractions of fiber type. Comment: 46 pages, minor revision, to appear in Advances in Mathematics |
| Databáze: | arXiv |
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