Cone of curves

Let X be a proper variety. By definition, a (real) 1-cycle on X is a formal linear combination C=\sum a_iC_i of irreducible, reduced and proper curves C_i, with coefficients a_i \in \mathbb{R}. Numerical equivalence of 1-cycles is defined by intersections: two 1-cycles C and C' are numerically equivalent if C \cdot D = C' \cdot D for every Cartier divisor D on X. Denote the real vector space of 1-cycles modulo numerical equivalence by N_1(X). We define the cone of curves of X to be : NE(X) = \left\{\sum a_i[C_i], \ 0 \leq a_i \in \mathbb{R} \right\} where the C_i are irreducible, reduced, proper curves on X, and [C_i] their classes in N_1(X). It is not difficult to see that NE(X) is indeed a convex cone in the sense of convex geometry. ==Applications==

One useful application of the notion of the cone of curves is the Kleiman condition, which says that a (Cartier) divisor D on a complete variety X is ample if and only if D \cdot x > 0 for any nonzero element x in \overline{NE(X)}, the closure of the cone of curves in the usual real topology. (In general, NE(X) need not be closed, so taking the closure here is important.) A more involved example is the role played by the cone of curves in the theory of minimal models of algebraic varieties. Briefly, the goal of that theory is as follows: given a (mildly singular) projective variety X, find a (mildly singular) variety X' which is birational to X, and whose canonical divisor K_{X'} is nef. The great breakthrough of the early 1980s (due to Mori and others) was to construct (at least morally) the necessary birational map from X to X' as a sequence of steps, each of which can be thought of as contraction of a K_X-negative extremal ray of NE(X). This process encounters difficulties, however, whose resolution necessitates the introduction of the flip. ==A structure theorem==

The above process of contractions could not proceed without the fundamental result on the structure of the cone of curves known as the Cone Theorem. The first version of this theorem, for smooth varieties, is due to Mori; it was later generalised to a larger class of varieties by Kawamata, Kollár, Reid, Shokurov, and others. Mori's version of the theorem is as follows: Cone Theorem. Let X be a smooth projective variety. Then 1. There are countably many rational curves C_i on X, satisfying 0, and : \overline{NE(X)} = \overline{NE(X)}_{K_X\geq 0} + \sum_i \mathbf{R}_{\geq0} [C_i]. 2. For any positive real number \epsilon and any ample divisor H, : \overline{NE(X)} = \overline{NE(X)}_{K_X+\epsilon H\geq0} + \sum \mathbf{R}_{\geq0} [C_i], where the sum in the last term is finite. The first assertion says that, in the closed half-space of N_1(X) where intersection with K_X is nonnegative, we know nothing, but in the complementary half-space, the cone is spanned by some countable collection of curves which are quite special: they are rational, and their 'degree' is bounded very tightly by the dimension of X. The second assertion then tells us more: it says that, away from the hyperplane \{C : K_X \cdot C = 0\}, extremal rays of the cone cannot accumulate. When X is a Fano variety, \overline{NE(X)}_{K_X\geq 0} = 0 because -K_X is ample. So the cone theorem shows that the cone of curves of a Fano variety is generated by rational curves. If in addition the variety X is defined over a field of characteristic 0, we have the following assertion, sometimes referred to as the Contraction Theorem: 3. Let F \subset \overline{NE(X)} be an extremal face of the cone of curves on which K_X is negative. Then there is a unique morphism \operatorname{cont}_F : X \rightarrow Z to a projective variety Z, such that (\operatorname{cont}_F)_* \mathcal{O}_X = \mathcal{O}_Z and an irreducible curve C in X is mapped to a point by \operatorname{cont}_F if and only if [C] \in F. (See also: contraction morphism). ==References==