Blown-up toric surfaces with non-polyhedral effective cone

We construct projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone. As a consequence, we prove that the pseudo-effective cone of the Grothendieck–Knudsen moduli space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mover accent="true"> <m:mi>M</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> M̄0,n of stable rational curves is not polyhedral for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>10</m:mn> </m:mrow> </m:math> n≥ 10 . These results hold both in characteristic 0 and in characteristic 𝑝, for all primes 𝑝. Many of these toric surfaces are related to an interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order. Our analysis 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.