Existence Of Minimal Models Research Articles

MMP for co-rank one foliations on threefolds

We prove existence of flips, special termination, the base point free theorem and, in the case of log general type, the existence of minimal models for F-dlt foliated pairs of co-rank one on a {mathbb {Q}}-factorial projective threefold. As applications, we show the existence of F-dlt modifications and F-terminalisations for foliated pairs and we show that foliations with canonical or F-dlt singularities admit non-dicritical singularities. Finally, we show abundance in the case of numerically trivial foliated pairs.

Minimisation and reduction of 5-coverings of elliptic curves

We consider models for genus one curves of degree 5, which arise in explicit 5-descent on elliptic curves. We prove a theorem on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and give an algorithm for computing such models. Finally we describe how to reduce genus one models of degree 5 defined over Q.

Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves

In this paper we consider models for genus one curves of degree n for n = 2, 3 and 4, which arise in explicit n-descent on elliptic curves. We prove theorems on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and provide simple algorithms for minimising a given model, valid over general number fields. Finally, for genus one models defined over Q, we develop a theory of reduction and again give explicit algorithms for n = 2, 3 and 4.