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 $\mathcal{X}\to S$.