On some Fano manifolds admitting a rational fibration

TL;DR

This paper investigates Fano manifolds with a specific invariant c_X=2, demonstrating that such manifolds, after certain modifications, admit a conic bundle structure or a Del Pezzo surface fibration, expanding understanding of their geometric structure.

Contribution

It characterizes Fano manifolds with c_X=2, showing they have a conic bundle or Del Pezzo fibration after birational modifications, extending previous classifications.

Findings

X has a conic bundle structure after flips

X admits an equidimensional Del Pezzo fibration

Weaker properties are shown for c_X=1

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.