Blown-up toric surfaces with non-polyhedral effective cone

Abstract

We construct examples of projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudoeffective cone, both in characteristic 0 and in prime characteristic. 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 an infinite set of primes of positive density. In particular, M 0,n is not a Mori Dream Space in characteristic p in this range, which is also a new result. Our analysis in characteristic p is based on applying tools of arithmetic geometry and Galois representations to a very interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order.

On intrinsic negative curves

“…where each C i is a (−2)-curve and a i , n ≥ 0. By [9,Cor. 3.17], the fact that res(C i ) = 0 for any i and the fact that Eff(Y ) is non-polyhedral we conclude that all the C i are contracted by π.…”

“…Thus res(C) is non-torsion and so [C] spans an extremal ray of Eff(X). By [9,Cor. 3.12] the divisor K X + C is linearly equivalent to an effective divisor whose support can be contracted.…”

“…We are now going to give an example of an infinite family of negative curves lying on the blowing-up at of a fixed toric surface. First of all we recall a construction from [9]. Given an expected lattice polygon ∆ ⊆ Q 2 of width m ∶= lw(∆), with vol(∆) = m 2 and ∂∆ ∩ Z 2 = m, if L ∆ (m) contains an unique irreducible element, then its strict transform C ⊆ X ∶= X ∆ is a curve of arithmetic genus one with C 2 = 0.…”

“…Its Mordell-Weil group Pic 0 (C)(Q) is free of rank one so that res(C) is either trivial or non-torsion. The first possibility is ruled out by the fact that dim C = dim L ∆ (6) = 0 and the exact sequence of the ideal sheaf of C in X (see [9,Lem. 3.2] for details).…”

The Fulton-MacPherson compactification is not a Mori dream space

“…Equally interesting seems to be the more delicate question of whether such a blowup has a polyhedral pseudoeffective cone (see e.g. [CLTU20], [LU21]). Our Proposition 3.2 and Corollary 3.3 provide a general method for finding such toric varieties via a series of certain ray contractions of toric fans, which extends a key result by A.-M. Castravet and J. Tevelev [CT15, Proposition 3.1].…”