Contractible classes in toric varieties

Abstract

Let X be a smooth, complete toric variety. Let A1(X) be the group of algebraic 1-cycles on X modulo numerical equivalence and N1(X) = A1(X)⊗ZQ . Consider inN1(X) the coneNE(X) generated by classes of curves on X . It is a well-known result due to M. Reid [13] that NE(X) is closed, polyhedral and generated by classes of invariant curves on X . The varietyX is projective if and only if NE(X) is strictly convex; in this case, a 1-dimensional face ofNE(X) is called an extremal ray. It is shown in [13] that every extremal ray admits a contraction to a projective toric variety. We think of A1(X) as a lattice in the Q -vector space N1(X). Suppose that X is projective. For every extremal ray R ⊂ NE(X), we choose the primitive class in R ∩ A1(X); we call this class an extremal class. The set E of extremal classes is a generating set for the cone NE(X), namely NE(X) = ∑ γ∈E Q≥0 γ. For many purposes it would be useful to have a linear decomposition with integral coefficients: for instance, what can we say about curves having minimal degree with respect to some ample line bundle onX? It is an open question whether extremal classes generate NE(X) ∩ A1(X) as a semigroup. In this paper we introduce a set C ⊇ E of classes in NE(X) ∩ A1(X) which is a set of generators of NE(X) ∩ A1(X) as a semigroup. Classes in C are geometrically characterized by “contractibility”: