Quartic Del Pezzo Surfaces over Function Fields of Curves

The geometry of spaces of rational curves of low degree on Fano threefolds is a very active area of algebraic geometry. One of the main goals is to understand the Abel-Jacobi morphism from the base of a space of such curves to the intermediate Jacobian of the threefold. For instance, does a given space of curves dominate the intermediate Jacobian? If so, what are the fibers of this morphism? Does it give the maximal rationally connected (MRC) quotient? Representative results in this direction are available for: • cubic threefolds [HRS05, JHS, IM00, MT01]; • Fano threefolds of genus six and degree 10 in P [DIM12]; • Fano threefolds of genus seven and degree 12 in P [IM07]; • moduli of vector bundles [Cas04]—this case makes clear that one cannot always expect the morphism to the intermediate Jacobian to give the MRC fibration. At the same time, del Pezzo fibrations π : X → P are equally interesting geometrically. Moreover, the special case where the rational curves happen to be sections of π is particularly important for arithmetic applications. It is a major open problem to determine whether or not sections exist over a non-closed ground field. Of course, the Tsen-Lang theorem gives sections when the ground field is algebraically closed. Even when there are rigid sections, these are typically defined over extensions of large degree. However, suppose that the space of sections of fixed height is rationally connected over the intermediate Jacobian IJ(X ). If the fibration is defined over a finite field and the space of sections descends to this field then π has sections over that field. Indeed, this follows by combining a theorem of Lang [Lan55] (principal homogeneous spaces for abelian varities over finite fields are trivial) and a theorem of Esnault