  We construct examples of projective toric surfaces whose blow-up at a general
point has a non-polyhedral pseudo-effective cone, both in characteristic $0$
and in every prime characteristic $p$. As a consequence, we prove that the
pseudo-effective cone of the Grothendieck-Knudsen moduli space $\overline
M_{0,n}$ of stable rational curves is not polyhedral for $n\geq 10$ in
characteristic $0$ and in characteristic $p$, for all primes $p$. Many of these
toric surfaces are related to a very interesting class of arithmetic threefolds
that we call arithmetic elliptic pairs of infinite order. Their analysis in
characteristic $p$ relies on tools of arithmetic geometry and Galois
representations in the spirit of the Lang-Trotter conjecture, producing toric
surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective
cone in characteristic $0$ and in characteristic $p$, for an infinite set of
primes $p$ of positive density.
