Numerical invariants of Fano 4-folds

Details Numerical invariants of Fano 4-folds Numerical invariants of Fano 4-folds

2012-01-26

arxiv.org/abs/1201.5464v1

Mathematics

Algebraic Geometry

11 pages

Scientific paper

Let X be a (smooth, complex) Fano 4-fold. For any prime divisor D in X, consider the image of N_1(D) in N_1(X) under the push-forward of 1-cycles, and let c_D be its codimension in N_1(X). We define an integral invariant c_X of X as the maximal c_D, where D varies among all prime divisors in X. One easily sees that c_X is at most rho_X-1 (where rho is the Picard number), and that c_X is greater or equal than rho_X-rho_D, for any prime divisor D in X. We know from previous works that if c_X > 2, then either X is a product of Del Pezzo surfaces and rho_X is at most 18, or c_X=3 and rho_X is at most 6. In this paper we show that if c_X=2, then rho_X is at most 12.

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