Blown-up toric surfaces with non-polyhedral effective cone

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