Quartic del Pezzo surfaces over function fields of curves

Brendan Hassett 1  and Yuri Tschinkel
  • 1 Department of Mathematics-MS136, Rice University, 6100 S. Main St., Houston, TX, 77251-1892, USA
  • 2 Courant Institute, New York University, 251 Mercer St., New York, NY, 10012, USA
  • 3 Simons Foundation, 160 Fifth Avenue, New York, NY, 10010, USA

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.

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