Algorithms for Projectivity and Extremal Classes of a Smooth Toric Variety

Abstract

In this paper we present two algorithms: the first tests the projectivity of a smooth complete toric variety and the second determines the extremal classes of the Mori cone of a smooth projective toric variety. The crucial fact is that we are able to give a complete description of $\aunox$, determining a basis $B$ of $\aunox$ and the coordinates with respect to $B$ of any element of $\aunox$. The computational condition testing the projectivity is obtained by Kleiman's criterion of ampleness, while the condition determining the extremality of a class comes directly from the definition of a nonextremal class. The algorithms are used to study the Mori cone of Fano toric $n$-folds with dimension $n\leq 4$ and Picard number $\rho \geq 3$, computing all extremal rays of the Mori cone. Moreover, we describe a toric almost Fano variety of dimension $3$ and Picard number $35$ together with its Mori cone.