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arXiv · 1307.5418

On some Fano manifolds admitting a rational fibration

C. Casagrande
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Abstract

Let X be a smooth, complex Fano variety. For every prime divisor D in X, we set c(D):=dim ker(r:H^2(X,R)->H^2(D,R)), where r is the natural restriction map, and we define an invariant of X as c_X:=max{c(D)|D is a prime divisor in X}. In a previous paper we showed that c_X<9, and that if c_X>2, then either X is a product, or X has a flat fibration in Del Pezzo surfaces. In this paper we study the case c_X=2. We show that up to a birational modification given by a sequence of flips, X has a conic bundle structure, or an equidimensional fibration in Del Pezzo surfaces. We also show a weaker property of X when c_X=1.

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C. Casagrande. 2013-07-20. On some Fano manifolds admitting a rational fibration. https://doi.org/10.1112/jlms%2Fjdu011

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