Quartic del Pezzo surfaces over function fields of curves

Abstract

We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.

Embedding pointed curves in K3 surfaces

“…The constructions in this paper have strong implications for the structure of spaces of sections for del Pezzo surface fibrations over P 1 . Details appear in [HT12].…”

On the moduli of degree 4 Del Pezzo surfaces

“…In Section 10 we state and prove our main theorems, describing and enumerating the components of general families of degree 4 Del Pezzo surfaces over P 1 . In an Appendix, we show that the discrete invariant of families introduced here agrees with the height defined in [18].…”

“…By Proposition 13, the degrees in (6.8) must be related by a constant proportionality. We deduce their equality from any of the worked out examples, e.g., Case 1 on page 11 of [18] with π * ω −1…”

“…We recall the construction from [18,Remark 15]. Let X ⊂ P 1 × P 5 be a complete intersection of a form of bidegree (1, 1) and two forms of bidegree (0, 2).…”

“…If we assume that family is generically smooth with square-free discriminant, then X is a smooth projective threefold. In this case in [18] the height is defined as a triple intersection number on X . Here we show that these two definitions agree.…”