2.12 Ample and nef cones
Here \(X\) is a projective variety over \(\mathbb {C}\).
The nef cone \(\operatorname{\mathrm{Nef}}X\) is the image in \(\operatorname{\mathrm{NS}}_{\mathbb {R}} X\) of the set of all nef \(\mathbf{R}\)-divisors. The ample cone \(\operatorname{\mathrm{Amp}}X\) is the image in \(\operatorname{\mathrm{NS}}_{\mathbb {R}} X\) of the set of all ample \(\mathbb {R}\)-divisors.
It is clear that \(\operatorname{\mathrm{Nef}}X\) and \(\operatorname{\mathrm{Amp}}X\) are both convex cones in the (finite-dimensional) real vector space \(\operatorname{\operatorname{\mathrm{NS}}_{\mathbb {R}}}X\). The reason is that they are images of convex cones under the \(\mathbb {R}\)-linear map \(\operatorname{\mathrm{Div}_{\mathbb {R}}}X\to \operatorname{\operatorname{\mathrm{NS}}_{\mathbb {R}}}X\), and it is easy to see that the image of a convex subset (resp. a cone) under a linear map is a convex subset (resp. a cone).
Suppose \(C\) is a convex cone in an vector space \(V\) with basis \(w_1,\ldots , w_n\) and \(v\in V\). Suppose further that \(v\pm w_i\in C\) for all \(i\). Then, for all \(y_1,\ldots , y_n\in [-1,1]\), \(nv+\sum y_i w_i\in C\). In particular, \(nv\) and \(v\) are both in the interior of \(C\).
We have
So, since \(y_i\in [0,1]\), \(v + y_i w_i\in C\). But then \(nv + \sum y_i w_i = \sum (v + y_i w_i)\in C\).
It follows directly that \(nv\) is in the interior of \(C\). But then, since the multipication by \(n\) map on \(V\) induces an isomorphism from \(C\) onto itself, it follows that \(v\) is in the interior of \(C\).
\(\operatorname{\mathrm{Nef}}X\) is closed, \(\operatorname{\mathrm{Amp}}X\) is open and \(\operatorname{\mathrm{Amp}}X\subseteq \operatorname{\mathrm{Nef}}X\).
It’s basically obvious that \(\operatorname{\mathrm{Amp}}X\subseteq \operatorname{\mathrm{Nef}}X\). To be pedantic about it, suppose \(D\in \operatorname{\mathrm{Amp}}X\). Then \(\deg D\cdot C {\gt} 0\) for all curves \(C\) by Nakai-Moishezon. But that directly implies that \(D\in \operatorname{\mathrm{Nef}}X\).
\(\operatorname{\mathrm{Nef}}X\) is closed because it is the intersection of closed half-spaces. To be more precise about this, let \(C\) be a curve on \(X\). Then we get an \(\mathbb {R}\)-linear functional \(\lambda _C:\operatorname{\operatorname{\mathrm{NS}}_{\mathbb {R}}}X\to \mathbb {R}\) given by \(D\mapsto \deg D\cdot C\). Then \(\operatorname{\mathrm{Nef}}X = \cap _{C} \lambda _C^{-1}([0,\infty ))\).
The openness of \(\operatorname{\mathrm{Amp}}X\) is a variation on Proposition 2.27. First, pick a basis \([E_1],\ldots , [E_{\rho }]\) of \(\operatorname{\operatorname{\mathrm{NS}}_{\mathbb {R}}}X\). We can do this with \(E_i\) integral for all \(i\) because \(\operatorname{\operatorname{\mathrm{NS}}_{\mathbb {R}}}X = \operatorname{\mathrm{NS}}X\otimes \mathbb {R}\). Suppose \([H]\in \operatorname{\mathrm{Amp}}X\) is represented by a \(H\in \operatorname{\mathrm{Div}_{\mathbb {R}}}X\). We can write \(H = \sum c_i H_i\) with \(c_i{\gt}0\) and \(H_i\) integral and ample. Then, the openness of ampleness of \(\mathbb {Q}\)-divisors, we can find a rational number \(\epsilon {\gt}0\) such that, \([H_1 \pm \epsilon E_j]\in \operatorname{\mathrm{Amp}}X\) for all \(j\). It follows then from Lemma 2.40 that \([H_1]\) is in the interior of \(\operatorname{\mathrm{Amp}}X\). But, since \(c_1{\gt}0\), that directly implies that \([H]\) is in the interior as well.
Finally, suppose \(D\in \operatorname{\mathrm{Amp}}X\). Then \(\deg D\cdot C {\gt} 0\) for all curves \(C\) by Nakai-Moishezon. But that directly implies that \(D\in \operatorname{\mathrm{Nef}}X\).
\(\overline{\operatorname{\mathrm{Amp}}X}\subseteq \operatorname{\mathrm{Nef}}X\) and \(\operatorname{\mathrm{Amp}}X\subseteq \operatorname{\mathrm{int}}\operatorname{\mathrm{Nef}}X\).
Obvious.
Kleiman showed that both of the containments in Corollary 2.43 are actually equalities of sets. But the proof requires the following.
Suppose \(D\in \operatorname{\mathrm{Nef}}X\) and \(V\) is a \(k\)-dimensional closed subvariety of \(X\). Then \(D^k\cdot V\geq 0\).
Proposition 2.28, which we already proved, is the special case of this for surfaces. In fact, the proof in arbitrary dimension, which you can find in [ 6 ] , is pretty similar to the special case. So let’s skip it.
Suppose \(D\) and \(H\) are \(\mathbb {R}\)-divisors.
If \(D\) is nef and \(H\) is ample, then \(D+xH\) is ample for all positive real \(x\).
If there exists \(\epsilon {\gt}0\) such that \(D+xH\) is ample for all \(x\) with \(0{\lt}x{\lt}\epsilon \), then \(D\) is nef.
a: Suppose \(D\) is nef and \(H\) is ample. We then have \(H = \sum _{i=1}^m c_i H_i\) with \(H_i\) ample and integral and \(c_i{\gt}0\). If we can show that \(D+xH_1\) is ample for all \(x{\gt}0\), then it will follow that \(D+xH\) is ample as \(D+xH\) is in the smallest convex cone containing \(D+xH_1\), \(H_2,\ldots , H_m\). So we can assume that \(H\) is integral. And we can even assume that \(H\) is very ample.
Then, by Bertini, for each \(s\geq 0\), \(H^s\cdot [V]\) is represented by an effective \((k-s)\) cycle and \(V\) is a \(k\)-dimensional subvariety. Then, if \(t\geq 0\) with \(s+t=k\), \(\int _V D^sH^k = D^t H^s\cdot V\geq 0\) by Theorem 2.44. It follows that
So, by Nakai-Moishezon, \(D+xH\) is ample.
b: Suppose the hypothesis of b holds and \(C\) is a curve. Then \((D+xH)\cdot C \geq 0\) for all sufficiently small positive \(x\). But then \(D\cdot C\geq 0\). So \(D\) is nef.
We have \(\operatorname{\mathrm{Amp}}X = \operatorname{\mathrm{int}}\operatorname{\mathrm{Nef}}X\) and \(\operatorname{\mathrm{Nef}}X=\overline{\operatorname{\mathrm{Amp}}X}\).
We already know that \(\operatorname{\mathrm{Amp}}X\subseteq \operatorname{\mathrm{int}}\operatorname{\mathrm{Nef}}X\). Suppose \(D\in \operatorname{\mathrm{int}}\operatorname{\mathrm{Nef}}X\) and \(H\in \operatorname{\mathrm{Amp}}X\). Then there exists \(\epsilon {\gt}0\) such that \(D+xH\in \operatorname{\mathrm{Nef}}X\) for all real \(x\) with \(|x|{\lt}\epsilon \). But then, setting \(D'= D-(\epsilon /2) H\), we get that \(D'\in \operatorname{\mathrm{Nef}}X\). So, by Theorem 2.45, we get that \(D' + xH\) is ample for all \(x{\gt}0\). In particular, \(D = D' + \epsilon H/2\) is ample.
Similarly, we already know that \(\overline{\operatorname{\mathrm{Amp}}X}\subseteq \operatorname{\mathrm{Nef}}X\). The reverse inclusion follows directly from Corollary 2.45a.