ScalingStacks

Now fix a ball $\mathcal{U}$ in $N^{1}(X)_{\mathbb{R}}$ centered at $\alpha$, such that $\mathcal{K}_{NS}\cap\mathcal{U}$ is defined by $\{\Phi_{\beta}>0\}_{1\leq\beta\leq k}$ where the $\Phi_{\beta}$ are linear forms with rational coefficients. Since the big cone is open, up to shrinking $\mathcal{U}$ we may also assume that all the classes in $\partial\mathcal{K}_{NS}\cap\mathcal{U}$ are big. We may add some more linear forms to the $\Phi_{\beta}$, until they define a strongly convex rational polyhedral cone $C$ which is contained in $\overline{\mathcal{K}}_{NS}\cap\mathcal{U}$. We can then write

$$C=\left\{\sum_{i=1}^{\ell}a_{i}\gamma_{i}\ \bigg|\ a_{i}\geq 0\right\},$$

where the $\gamma_{i}$ are nef and big classes in $\mathcal{U}$. We claim that, when $t$ is bigger than some $t_{0}<1$, it is possible to write the path $\alpha_{t}$ as $\sum_{i}a_{i}(t)\gamma_{i}$ where the functions $a_{i}(t)$ are continuous and nonnegative. Assume first that the cone $C$ is simplicial, which means that the $\gamma_{i}$ are linearly independent. Then the path $\alpha_{t}$ enters and eventually stays in $C$, and so it can be expressed uniquely as

(4.1)

$$\alpha_{t}=\sum_{i=1}^{\ell}a_{i}(t)\gamma_{i},$$

where the $a_{i}(t)$ are smooth and nonnegative, $t_{0}\leq t\leq 1$. If on the other hand $C$ is not simplicial, it can be written as a finite union of simplicial subcones that intersect only along faces, and that are spanned by some linearly independent subsets of the $\gamma_{i}$. On any time interval when $\alpha_{t}$ belongs to the interior of a simplicial cone, the coefficients $a_{i}(t)$ in (4.1) vary smoothly, and on a common face of two simplicial cones the coefficients agree, hence the $a_{i}(t)$ vary continuously when $t_{0}\leq t<1$. Moreover since we only have finitely many simplicial subcones, we see that as $t\to 1$ the $a_{i}(t)$ converge to the coefficients of $\alpha_{1}$ in any of the simplicial cones that contains it, and so the $a_{i}(t)$ are continuous on the whole interval $t_{0}\leq t\leq 1$.