For $l\in{\cal L}$ we will define an order function

$$ord_{l}\in\Gamma((0,+\infty),f_{l}^{\ast}(Aff_{{\bf Z},Y}))$$

(its meaning will become clear later) by the following inductive procedure:

1. 1.

   Let $l\in{\cal L}_{in}$ and $t>0$ be sufficiently small. Then in the standard affine coordinates near $s=f_{l}(0)$ one has $\alpha_{l}=\pm f_{l}^{\ast}(dy)$. We define $ord_{l}=\pm f_{l}^{\ast}(y)$. Then $d(ord_{l})=\alpha_{l}$, and we can extend uniquely $ord_{l}$ for all $t\in(0,+\infty)$.

2. 2.

   Let $l\in{\cal L}_{com}$ and $l_{1},l_{2}$ be parents of $l$. In the notation of Axiom 3 we have $f_{l_{1}}(t_{1})=f_{l_{2}}(t_{2})=f_{l}(0)$ and $\alpha_{l}(0)=n_{1}\alpha_{l_{1}}(t_{1})+n_{2}\alpha_{l_{2}}(t_{2})$. Then we define $ord_{l}(0):=n_{1}ord_{l_{1}}(t_{1})+n_{2}ord_{l_{2}}(t_{2})$. Again, using the condition $d(ord_{l})=\alpha_{l}$ and the knowledge of $ord_{l}(0)$ we can extend $ord_{l}$ for $t>0$.

Notice that $ord_{l}(t)$ can be thought of as affine function on the tangent space $T_{f_{l}(t)}Y$ (in the induced integral affine structure). In particular, we have a half-plane $P_{l,t}\subset T_{f_{l}(t)}Y$ defined by the inequality $ord_{l}(t)>0$. The family of half-planes $P_{l,t}$ is covariantly constant with respect to $\nabla^{aff}$.

Each half-plane $P_{l,t}$ contains $0\in T_{f_{l}(t)}Y$ strictly in its interior. Recall that at the end of Section 9.1 we defined another half-plane $P_{l,t}^{(0)}\subset T_{f_{l}(t)}Y$. It is easy to see that $P_{l,t}^{(0)}$ is the half-plane parallel to $P_{l,t}$ such that $0\in T_{f_{l}(t)}Y$ is on the boundary of $P_{l,t}^{(0)}$.