1.2. Numerical classes and positivity [01DS]

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1.2. Numerical classes and positivity

Let $\mathcal{X}$ be a normal projective $S$ -variety.

Lemma 1.2.

Assume that $\mathcal{L}\in\Pic(\mathcal{X})$ is nef on $\mathcal{X}_{0}$ , i.e. $\mathcal{L}\cdot C\geq 0$ for all $k$ -proper curves $C$ in $\mathcal{X}_{0}$ . Then $\mathcal{L}$ is also nef on $\mathcal{X}_{K}$ , i.e. $\mathcal{L}\cdot C\geq 0$ for all $K$ -proper curves $C$ of $\mathcal{X}_{K}$ as well.

We will then simply say that $\mathcal{L}$ is nef.

Proof.

Let $C$ be $K$ -proper curve in $\mathcal{X}_{K}$ and let $\mathcal{C}$ be its closure in $\mathcal{X}$ . Since $\mathcal{C}$ is flat over $S$ , the degree of $\mathcal{L}|_{\mathcal{C}}$ on the generic fiber and on the special fiber concide, which reads $\mathcal{L}\cdot C=\mathcal{L}\cdot\mathcal{C}_{0}$ . Now $\mathcal{C}_{0}$ is an effective linear combination of vertical curves, and the result follows. ∎

We recall the following standard notions.

Definition 1.3.

Let $\mathcal{X}$ be a normal projective $S$ -variety as above.

(i)

The space $N^{1}(\mathcal{X}/S)$ of codimension 1 numerical classes is defined as the quotient of $\Pic(\mathcal{X})_{\mathbf{R}}$ by the subspace spanned by numerically trivial line bundles, i.e. those $\mathcal{L}\in\Pic(\mathcal{X})$ such that $\mathcal{L}\cdot C=0$ for all projective curves contained in a fiber of $\mathcal{X}\to S$ .
(ii)

The nef cone $\Nef(\mathcal{X}/S)\subset N^{1}(\mathcal{X}/S)$ is defined as the set of numerical classes $\alpha\in N^{1}(\mathcal{X}/S)$ such that $\alpha\cdot C\geq 0$ for all projective curves contained in a fiber of $\mathcal{X}\to S$ .

Note that the $\mathbf{R}$ -vector space $N^{1}(\mathcal{X}/S)$ is finite dimensional. Indeed Lemma 1.2 shows that the restriction map $N^{1}(\mathcal{X}/S)\to N^{1}(\mathcal{X}_{0}/k)$ is injective, and the latter space is finite dimensional since $\mathcal{X}_{0}$ is projective over $k$ . Observe also that $\Nef(\mathcal{X}/S)$ is a closed convex cone of $N^{1}(\mathcal{X}/S)$ . Lemma 1.2 implies that $\Nef(\mathcal{X}/S)=\Nef(\mathcal{X}_{0}/k)\cap N^{1}(\mathcal{X}/S)$ under the injection $N^{1}(\mathcal{X}/S)\to N^{1}(\mathcal{X}_{0}/k)$ .

We have the following standard fact:

Lemma 1.4.

Let $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ be a vertical blow-up.

(i)

There exists a $\pi$ -ample divisor $A\in\Div_{0}(\mathcal{X}^{\prime})$ .
(ii)

If $\mathcal{L}\in\Pic(\mathcal{X})$ is ample then there exists $m\in\mathbf{N}$ such that $\pi^{*}(m\mathcal{L}|_{\mathcal{X}_{K}})$ extends to an ample line bundle $\mathcal{L}^{\prime}$ on $\mathcal{X}^{\prime}$ .

Proof.

By definition, there exists a vertical ideal sheaf $\mathfrak{a}$ on $\mathcal{X}$ such that $\pi$ is obtained as the blow-up of $\mathcal{X}$ along $\mathfrak{a}$ . The universal property of blow-ups yields a $\pi$ -ample Cartier divisor $A$ on $\mathcal{X}^{\prime}$ such that $\mathfrak{a}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(A)$ , and $A$ is also vertical since $\mathfrak{a}$ is, which proves (i). If $\mathcal{L}$ is ample on $\mathcal{X}$ then $m\pi^{*}\mathcal{L}+A$ is ample on $\mathcal{X}^{\prime}$ for $m\gg 1$ , and (ii) follows. ∎

Recall that an $\mathbf{R}$ -line bundle on $\mathcal{X}$ (resp. $\mathcal{X}_{K}$ ) is ample if it can be written as a positive linear combination of ample line bundles. As a direct consequence of Lemma 1.4 we get:

Corollary 1.5.

If $L\in\Pic(\mathcal{X}_{K})_{\mathbf{R}}$ is ample then $L$ extends to an ample $\mathbf{R}$ -line bundle $\mathcal{L}\in\Pic(\mathcal{X}^{\prime})_{\mathbf{R}}$ for all sufficiently high vertical blow-ups $\mathcal{X}^{\prime}\to\mathcal{X}$ .

We shall use the following version of the Negativity Lemma, cf. [KM98, Lemma 3.39]. The proof we give is a variant of the argument used in [BdFF10, Proposition 2.11].

Lemma 1.6.

Assume that $\mathcal{X}$ is vertically $\mathbf{Q}$ -factorial and let $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ be a vertical blow-up. If $D\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{R}}$ is $\pi$ -nef then $\pi^{*}\pi_{*}D-D$ is effective.

Proof.

As a first step, we reduce the assertion to the case where $D\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$ is $\pi$ -ample. Indeed, the set of vertical $\pi$ -ample $\mathbf{R}$ -divisors, which is an open convex cone in $\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{R}}$ , is non-empty by (i) of Lemma 1.4. We may thus choose a basis $A_{1},...,A_{r}$ of $\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{R}}$ made up of $\pi$ -ample Cartier divisors. Let $\varepsilon=(\varepsilon_{i})\in\mathbf{R}_{+}^{r}$ be such that $D_{\varepsilon}:=D+\sum_{i}\varepsilon_{i}A_{i}$ is a $\mathbf{Q}$ -divisor. The fact that $D$ is $\pi$ -nef means that $D\cdot C\geq 0$ for each curve $C$ contained in a fiber of $\pi$ . Since each $A_{i}$ is $\pi$ -ample, it follows from Kleiman’s criterion [Kle66] that $D_{\varepsilon}$ is $\pi$ -ample on the projective $k$ -scheme $\mathcal{X}_{0}^{\prime}$ , hence $D_{\varepsilon}$ is also $\pi$ -ample on $\mathcal{X}^{\prime}$ by [EGA, III.4.7.1]. Upon replacing $D$ with $D_{\varepsilon}$ for $\varepsilon$ arbitrarily small we may thus assume as desired that $D\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$ is $\pi$ -ample.

Now choose $m\gg 1$ such that $\mathcal{O}_{\mathcal{X}^{\prime}}(mD)$ is $\pi$ -globally generated, which means that the vertical fractional ideal sheaf $\mathfrak{a}:=\pi_{*}\mathcal{O}_{\mathcal{X}^{\prime}}(mD)$ satisfies $\mathfrak{a}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(mD)$ . It is obvious that $\mathfrak{a}\subset\mathcal{O}_{\mathcal{X}}(m\pi_{*}D)$ , hence $\mathcal{O}_{\mathcal{X}^{\prime}}(mD)\subset\mathcal{O}_{\mathcal{X}^{\prime}}(m\pi^{*}\pi_{*}D)$ , and the result follows. ∎