Hilbert’s 14th problem over finite fields and a conjecture on the cone of curves

Abstract

We give the first examples over finite fields of rings of invariants that are not finitely generated. (The examples work over arbitrary fields, for example the rational numbers.) The group involved can be as small as three copies of the additive group. The failure of finite generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell-Weil rank. Our work suggests a generalization of the MorrisonKawamata cone conjecture on Calabi-Yau fiber spaces to klt Calabi-Yau pairs. We prove the conjecture in dimension two under the assumption that the anticanonical bundle is semi-ample.

Big rational surfaces

“…Then −K X is effective and big if and only if one of the following holds: There are smooth projective rational surfaces with finitely generated Cox ring, whose anticanonical divisor is not big. For example, by [26,Theorem 5.2] the surface X obtained by blowing up the nine inflection points of a smooth plane cubic has finitely generated Cox ring. However, the anticanonical divisor −K X is not big since |−K X | contains an irreducible curve and K 2 X = 0.…”

A special Cayley octad

“…Our example therefore gives one of the few pieces of evidence for this broader version of the conjecture. (The other main example I am aware of is Totaro's proof of the conjecture for rational elliptic surfaces [Tot,Theorem 8.2]. )…”

Cremona special sets of points in products of projective spaces

“…We assume that all fibres are irreducible which is equivalent to the condition that the set P is unnodal. When m = 1 this can be achieved by assuming that no three points are collinear ( [24], Lemma 3.1). Let C be the unique cubic curve passing through the base points p 1 , .…”

“…This action defines a birational action on X, and, the condition of minimality implies that the action embeds A(K) into the group Aut ps (X) of pseudo-automorphisms of X [20], [24], Lemma 6.2.…”