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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.
We give examples of failure of the existence of co-fibered products in the category of algebraic curves.
We study applications of divisibility properties of recurrence sequences to Tate’s theory of abelian varieties over finite fields.
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