On The Geometry Of Elliptic Pairs

An elliptic pair $(X, C)$ is a projective rational surface $X$ with log terminal singularities, and an irreducible curve $C$ contained in the smooth locus of $X$, with arithmetic genus one and self-intersection zero. They are a useful tool for determining whether the pseudo-effective cone of $X$ is polyhedral, and interesting algebraic and geometric objects in their own right. Especially of interest are toric elliptic pairs, where $X$ is the blow-up of a projective toric surface at the identity element of the torus. In this paper, we classify all toric elliptic pairs of Picard number two. Strikingly, it turns out that there are only three of these. Furthermore, we study a class of non-toric elliptic pairs coming from the blow-up of $\mathbb{P}^2$ at nine points on a nodal cubic, in characteristic $p$. This construction gives us examples of surfaces where the pseudo-effective cone is non-polyhedral for a set of primes $p$ of positive density, and, assuming the generalized Riemann hypothesis, polyhedral for a set of primes $p$ of positive density.