Kleiman's and Nakai-Moishezon's ampleness criteria

Question

I would like to work out a simple example to understand the relation between Kleiman ampleness criterion and Nakai-Moishezon ampleness criterion.

Namely, let $X$ be the blow-up of $\mathbb{P}^{2}$ at two points $p,q$, $L$ the pull-back of a general line in $\mathbb{P}^{2}$, $R$ the strict transform of $\left\langle p,q\right\rangle$, and $E_1,E_2$ the exceptional divisors. We consider the divisor $D = 2L-E_1$ (i.e. pull-back of a conic through $p$).

1. (Nakai-Moishezon) We can write a curve $C\subset X$ as $C = dL-aE_1-bE_2$ (that is $C$ is the strict transform of a curve of degree $d$ in $\mathbb{P}^{2}$ with a point of multiplicity $a$ in p and a point of multiplicity $b$ in $q$). Now, $D\cdot C = 2d-a$. Since $a< d$ we get $D\cdot C > 0$. Moreover $D^{2} = 3 > 0$. Therefore, by Nakai-Moishezon criterion, $D$ is ample.
2. (Kleiman) The cone of curves $NE(X)$ of $X$ is generated by $R,E_1,E_2$. Let us consider $C = aR+bE_1+cE_2\in NE(X)$. Then $a,b,c$ are non-negative real numbers. We have $D\cdot C = a+b > 0$. To conclude using Kleiman criterion we should verify this also on the clousure of $NE(X)$. In this case $NE(X)$ is close. However, in general it seems to me that the condition on the closure of $NE(X)$ corresponds to $D^2 >0$ in Nakai-Moishezon criterion. Is this correct? On the other hand it seems that to use Kleiman criterion on just on the interior of the cone of curves one should prove that $D\cdot C > \delta$ where $\delta$ is a positive real number. Is this true? For instance in our case $D\cdot C = a+b$ where $a,b$ are non-negative real numbers (at least one different from zero). If the cone $NE(X)$ is not closed on the boundary we may have $D\cdot C = 0$.

Now, we consider $\mathbb{P}^3$ blown-up in a point $p$ and $D = 2H-E_1$. If $L$ is the strict transform of a line and $R$ a line in $E_1$ a curve in $X = Bl_p(\mathbb{P}^3)$ can be written as $C = dL-aR$. We have $D\cdot C = 2d-a > 0$. Is there a condition (like $D^2>0$ in Nakai-Moishezon criterion) ensuring that $D$ is ample? If not how can we apply Kleiman criterion if we do not know that $NE(X)$ is closed? Is there any result giving conditions for $NE(X)$ to be closed?

Answer 1

1. The Nakai--Moishezon criterion applies in any dimension; the condition is that $D^k \cdot C >0$ for any $k$-dimensional subvariety $C$ and $D^{\text{ dim } X}>0$.

2. There are several sufficient conditions for $NE(X)$ to be closed, e.g. if $X$ is Fano or if $X$ is toric (both true in your example). Much more simply, your $X$ has Picard number 2, and we can see the extremal rays of $NE(X)$ by inspection: namely the class of a line in $E_1$, and the class of the proper transform of a line through $p$. (They must be on the boundary of $NE(X)$, since the nef line bundle $H$ has degree 0 on the first one, while the nef line bundle $H-E_1$ has degree 0 on the second.)