Quasi elementary contractions of Fano manifolds

نویسنده

  • Cinzia Casagrande
چکیده

Smooth complex Fano varieties of dimension n form a limited family, so they have only a finite number of possible topological types. However not much is known about their topological invariants in higher dimensions. In particular we consider here the second Betti number b2 of a smooth Fano variety X, which coincides with the Picard number ρX . Recall that a Del Pezzo surface S has ρS ≤ 9. Fano 3-folds have been classified by Iskovskikh, Mori, and Mukai, see [IP99] and references therein. Thus we know that a Fano 3-fold X has ρX ≤ 10. In fact, more is true: as soon as ρX ≥ 6, X is a product of a Del Pezzo surface with P1 [MM81, Theorem 2]. Starting from dimension 4, we do not have a bound on ρX . The known examples with largest Picard number are just products of Del Pezzo surfaces with Picard number 9, which gives ρX = 9 2n. Optimistically one could think that Fano varieties with large Picard number are simpler, maybe a product of lower dimensional varieties. This would yield a linear bound (in the dimension n) for ρX , in fact one could expect precisely ρX ≤ 9 2n (see [Deb03, p. 122]). This is actually what happens in the toric case: if X is a smooth toric Fano variety of dimension n, then ρX ≤ 2n, and equality holds if and only if n is even and X is (S) n

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تاریخ انتشار 2007