First in Mori's cone

In 1966, Kleimann's paper ([Klei66]) was published on Annals of math, and it mostly determined a new kind of technique which would be used widely after that in studying birational geometry, especially by Mori. He (Kleimann) constructed a space of 1-cycle under the numerical relation generated by curves which isomorphics to $\mathbb{R}^n$ with usual topology and that gave us chances to study algebraic geometry in the light of euclidian space. Both Kleimann and Mori used this to study about the set of positive multiplicity 1-cycle $NE(X) = \sum_{C \subset X} \mathbb{R}_{\geq 0} [C]$ and its closure in  $\mathbb{R}^n$ in usual topology. The $\overline{NE(X)}$ is also called Kleiman-Mori cone [Mum-Oda]. Kleimann provided a criterion of ampleness by using $\overline{NE(X)}$. Mori even went futher. He pointed out the relation between rational curves, ample invertible sheaf and $\overline{NE(X)}$ by a theorem which is called "cone theorem" in [Mor82].In this post, I will give some properties and lemmas with more detailed proofs and comments of $\overline{NE(X)}$ in [Kol-Mor98].

First, we will come to a theorem about ampleness criterion.

proposition 1: The intersection number $(L.Z)$ define a perfect pairing:
                                                   $(Pic(X)/ \div) \times (Z_1(X)/ \div) \rightarrow \mathbb{Z}$
And these two are finite dimensional with dimension $\rho = \rho(X)$

Theorem 1: Let $X$ be a projective variety and $D$ is a Cartier divisor on $X$. then $D$ is ample iff
                                                   $\lbrace z | (D.z) > 0 \rbrace \supset \overline{NE(X)} -  0$ for all $z \in \overline{NE(X)} -  0$

Before proving this theorem, we should know about the key theorem for the proof.

Theorem 2: Let $X$ be the proper variety and $L$ a nef Cartier divisor. Then $(L^{dim Z}.Z) \geq 0$ for every integral closed subscheme $Z \subset X$.

proof of theorem 1. Assume $D$ is ample. By propostition 1, $N_1^{*} \otimes Q$  is a $\rho$-dimension space. Hence, we can consider $D$ as an element of basis $D_1=D, D_2, ..., D_{\rho}$. There is a lemma in [Laz04] that for any divisor $M$ and ample divisor $H$, then there exist a small enough $\epsilon > 0$ s.t $H+\epsilon M$ is ample. And we can also choose $\epsilon'$ s.t $H - \epsilon' M$ is ample and then $H + min(\epsilon, \epsilon')$ and $H- min(\epsilon, \epsilon')$ is ample. Apply to our basis, we can get a new basis $D, D+t_2D_2= D'_1,... D+t_{\rho}D_{\rho}=D'_{\rho}$ s.t all is ample and $ 2D - D_i$ is ample for each $i$. If there is some $C \in N_1$ s.t $(C.D_i) = 0$ for all $i$, then $(C.L) = 0$ for all $L \in Pic (X)$ which means $C = 0$. So $n: N_1 \rightarrow \mathbb{R}$, $n(C) = \sum |(C.D_i)|$ must be a norm.  Then $(2\rho D.C) - n(C) \geq \sum(2D-D_i.C) \geq 0$ because ample divisor is nef. Hence, $(D.C) \geq \frac{n(C)}{2\rho} > 0$ for all $C \in   \overline{NE(X)} -  0$.
Conversly, by lemma 4.2.1 [Klei66], any elements in $NE(X)$ can be written in the form $\sum_{q=1}^{\rho} a_q X_q$ with $a_q \geq 0$. Because $(D.z) > 0$ for all $z \in \overline{NE(X)} -  0$, fix an ample sheaf $A$, then there exist $t >>0$ such that $(tD - A..X_q) > 0$ for all $q$, and hence $(tD-A.Z) > 0$ for all $Z$ integral curve ($[Z] \in NE(X)$ then $[Z] =\sum_{q=1}^{\rho} a_q X_q$ with $a_q \geq 0$ ). By Theorem 2, $tD = A+L$ s.t $(tD^{dim Z}.Z) \geq (A^{dim Z}.Z) > 0$, and hence $(D^{dim Z}.Z) > 0$. Thus $D$ is ample.

Conferences:

 [Laz04] Robert Lazarsfeld. Positivity in algebraic geometry. I, volume 48 of Ergebnisse der
Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 2004. Classical setting:
line bundles and linear series.
 [Klei66] Steven L. Kleiman. Toward a Numerical Theory of Ampleness. Annals of Mathematics, Second Series, Vol. 84, No. 3 (Nov., 1966), pp. 293-344.
 [Mum-Oda] Mumford, Oda. Algebraic geometry II. 
 

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