Our main result is the description of generators of the total coordinate ring of the blow-up of P n in any number of points that lie on a rational normal curve. As a corollary we show that the algebra of invariants of the action of a two-dimensional vector group introduced by Nagata is finitely generated by certain explicit determinants. We also prove the finite generation of the algebras of invariants of actions of vector groups related to T-shaped Dynkin diagrams introduced by Mukai.

ABSTRACT. We give a conjectural description for the cone of effective divisors of the Grothendieck-Knudsen moduli space M 0,n of stable rational curves with n marked points. Namely, we introduce new combinatorial structures called hypertrees and show that they give exceptional divisors on M 0,n with many remarkable properties.

Abstract. Let C be a smooth complex projective curve of genus g ≥ 2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1. For any k ≥ 2, we find two irreducible components of the space of rational curves of degree k on M , both of the expected dimension. One component, which we call the nice component, has the property that when k is sufficiently large, the general element of M is a very free curve. The second component, that we call the almost nice component, has the property that the general element is a free curve f :for some positive vector bundle A on P 1 . We prove that the maximal rationally connected quotient of each component is the Jacobian J(C) of the curve C.

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.