Cone of curves

In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety formula_1 is a combinatorial invariant of importance to the birational geometry of formula_1. 
Definition.
Let formula_1 be a proper variety. By definition, a (real) "1-cycle" on formula_1 is a formal linear combination formula_5 of irreducible, reduced and proper curves formula_6, with coefficients formula_7. "Numerical equivalence" of 1-cycles is defined by intersections: two 1-cycles formula_8 and formula_9 are numerically equivalent if formula_10 for every Cartier divisor formula_11 on formula_1. Denote the real vector space of 1-cycles modulo numerical equivalence by formula_13.
We define the "cone of curves" of formula_1 to be
where the formula_6 are irreducible, reduced, proper curves on formula_1, and formula_18 their classes in formula_13. It is not difficult to see that formula_20 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 formula_11 on a complete variety formula_1 is ample if and only if formula_23 for any nonzero element formula_24 in formula_25, the closure of the cone of curves in the usual real topology. (In general, formula_20 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 formula_1, find a (mildly singular) variety formula_28 which is birational to formula_1, and whose canonical divisor formula_30 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 formula_1 to formula_28 as a sequence of steps, each of which can be thought of as contraction of a formula_33-negative extremal ray of formula_20. 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 Kollár, Reid, Shokurov, and others. Mori's version of the theorem is as follows:
Cone Theorem. Let formula_1 be a smooth projective variety. Then
1. There are countably many rational curves formula_6 on formula_1, satisfying formula_38, and 
2. For any positive real number formula_40 and any ample divisor formula_41,
where the sum in the last term is finite.
The first assertion says that, in the closed half-space of formula_13 where intersection with formula_33 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 formula_1. The second assertion then tells us more: it says that, away from the hyperplane formula_46, extremal rays of the cone cannot accumulate. When formula_1 is a Fano variety, formula_48 because formula_49 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 formula_1 is defined over a field of characteristic 0, we have the following assertion, sometimes referred to as the Contraction Theorem:
3. Let formula_51 be an extremal face of the cone of curves on which formula_33 is negative. Then there is a unique morphism formula_53 to a projective variety "Z", such that formula_54 and an irreducible curve formula_8 in formula_1 is mapped to a point by formula_57 if and only if formula_58.
(See also: contraction morphism).