# Cone of curves

## Definition

$NE(X)=\left\{\sum a_{i}[C_{i}],\ 0\leq a_{i}\in \mathbb {R} \right\}$ ## 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 Kollár, Reid, Shokurov, and others. Mori's version of the theorem is as follows:

${\overline {NE(X)}}={\overline {NE(X)}}_{K_{X}\geq 0}+\sum _{i}\mathbf {R} _{\geq 0}[C_{i}].$ ${\overline {NE(X)}}={\overline {NE(X)}}_{K_{X}+\epsilon H\geq 0}+\sum \mathbf {R} _{\geq 0}[C_{i}],$ 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.

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: