# Cone of curves

In mathematics, the **cone of curves** (sometimes the **Kleiman-Mori** cone) of an algebraic variety
is a combinatorial invariant of importance to the birational geometry of
.

## Definition

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.

## Applications

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

1. There are countably many rational curves on , satisfying , and

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
.
(See also: contraction morphism).