Contractible classes in toric varieties

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) ⊗Z Q . Consider in N1(X) the cone NE(X) generated by classes of curves on X. It is a well-known result due to M. Reid [11] that NE(X) is closed, polyhedral and generated by classes of invariant curves on X. The variety X is projective if and only if NE(X) is strictly convex; in this case, a 1-dimensional face of NE(X) is called an extremal ray. It is shown in [11] 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 on X? 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”: Definition 2.3. Let γ ∈ NE(X) ∩ A1(X) be primitive along A1(X) ∩ Q≥0 γ. We say that γ is contractible if there exist a toric variety Xγ and an equivariant morphism φγ : X → Xγ , with connected fibers, such that for every curve C in X φγ(C) = {pt} if and only if [C] ∈ Q≥0γ. This definition does not need the projectivity of X. We give a combinatorial characterization of contractibility in terms of the fan of X, and we show that a class γ is contractible if and only if every irreducible invariant curve in the class is extremal in every irreducible invariant surface containing it (theorem 2.2). In the projective case, this property is false for extremal classes: it can happen that every invariant curve in a class is extremal in every invariant subvariety containing it, but the class is not extremal in X (see example on page 7). When X is projective, all extremal classes are contractible, and a contractible class γ is extremal if and only if Xγ is projective. Hence, contractible non-extremal classes correspond to birational contractions to nonprojective toric varieties (corollary 3.5). Moreover, we show that a class γ ∈ NE(X) is contractible if and only if it is extremal in the subvariety A given by the intersection of all irreducible invariant divisors having negative intersection with γ.