Title: Blown up toric surfaces with non-polyhedral effective cone

Abstract: I will discuss joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia on constructing 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 of stable rational curves with n markings is not polyhedral for n>=10. Many of these toric surfaces are related to an interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order.