Blown-up toric surfaces with non-polyhedral effective cone

2020/09/29 by Castravet, Ana-Maria, Laface, Antonio, Tevelev, Jenia +1 · 1 citation
#14C20 #14E30 #14H10 #14H52 #14M25 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2009.14298

Abstract

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 M_0,n of stable rational curves is not polyhedral for n≥ 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.

Cited by

• Isoresidual fibrations and the birational geometry of moduli of pointed stable curves

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