The Geometry of Elliptic Pairs

Elliptic pairs consist of a surface X and a curve C on X satisfying properties similar to an elliptic curve. They are a useful tool for understanding the cone of effective divisors of X, and interesting geometric objects in their own right.

In this talk we will classify elliptic pairs where the surface X is toric and comes from a triangle. Furthermore, we study a class of non-toric elliptic pairs coming from the blow-up of the projective plane at nine points on a nodal cubic, over a finite field. This construction gives us examples of surfaces where the pseudo-effective cone is non-polyhedral for a set of primes of positive density, and, assuming the generalized Riemann hypothesis, polyhedral for a set of primes of positive density.