Cone of curves
Let be a proper variety. By definition, a (real) 1-cycle on is a formal linear combination of irreducible, reduced and proper curves , with coefficients . Numerical equivalence of 1-cycles is defined by intersections: two 1-cycles and are numerically equivalent if for every Cartier divisor on . Denote the real vector space of 1-cycles modulo numerical equivalence by .
We define the cone of curves of to be
where the are irreducible, reduced, proper curves on , and their classes in . It is not difficult to see that is indeed a convex cone in the sense of convex geometry.
One useful application of the notion of the cone of curves is the Kleiman condition, which says that a (Cartier) divisor on a complete variety is ample if and only if for any nonzero element in , the closure of the cone of curves in the usual real topology. (In general, 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 , find a (mildly singular) variety which is birational to , and whose canonical divisor 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 to as a sequence of steps, each of which can be thought of as contraction of a -negative extremal ray of . 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 be a smooth projective variety. Then
2. For any positive real number and any ample divisor ,
where the sum in the last term is finite.
The first assertion says that, in the closed half-space of where intersection with 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 . The second assertion then tells us more: it says that, away from the hyperplane , extremal rays of the cone cannot accumulate.
If in addition the variety is defined over a field of characteristic 0, we have the following assertion, sometimes referred to as the Contraction Theorem:
3. Let be an extremal face of the cone of curves on which is negative. Then there is a unique morphism to a projective variety Z, such that and an irreducible curve in is mapped to a point by if and only if .