We write $b=\min_{i\leqslant n}b_{i}$; we assume $b$ to be negative or zero by the condition $b_{1}+\ldots+b_{n}=2$, as the case $n=2$ and $b_{1}=b_{2}=1$ is already treated in the proof of [NXY19, prop. 5.4].

The blow-up $\mathscr{X}_{1}$ of the point $p_{\infty}$ in $\mathscr{X}$ yields a new irreducible component $D_{\infty,1}$ (we denote the strict transforms by the same letters for notational simplicity) with multiplicity $N_{\infty,1}=n+1$, the point $p_{\infty,1}=C\cap D_{\infty,1}$ and the intersection numbers $b_{i,1}\coloneqq-(C\cdot D_{i})_{\mathscr{X}_{1}}=b_{i}+1$. If we repeat the process $s$ times, we obtain the models $\mathscr{X}_{s}$, the exceptional divisors $D_{\infty,s}$ with multiplicity $N_{\infty,s}=ns+1$, the points $p_{\infty,s}=C\cap D_{\infty,s}$ and the intersection numbers $b_{i,s}\coloneqq-(C\cdot D_{i})_{\mathscr{X}_{s}}=b_{i}+s$.

For $s=1-b$, we have $\min_{i\leqslant n}\{b_{i,1-b}\}>0$, and by [NXY19] the integral affine structure induced by $\mathscr{X}_{1-b}$ on $\Star(\tau_{C})$ is given by $v_{0},\ldots,v_{n}$ and

(3.1.2)

$$v_{\infty,1-b}=\frac{1}{n(1-b)+1}(-1,b_{1}+1-b,\ldots,b_{n-1}+1-b).$$

The sequence of blow-ups $\mathscr{X}_{s+1}\rightarrow\mathscr{X}_{s}$ induces (weighted) barycentric subdivisions of the faces $\tau_{p_{\infty,s}}$ with vertices such that

(3.1.3)

$$N_{\infty,s+1}v_{\infty,s+1}=N_{\infty,s}v_{\infty,s}+\sum_{i=1}^{n}v_{i}.$$

Combining Eq. 3.1.2 and Eq. 3.1.3, at each step we obtain that

$$v_{\infty,s}=\frac{1}{ns+1}(-1,b_{1}+s,\ldots,b_{n-1}+s),$$

and in particular $v_{\infty}=(-1,b_{1},\ldots,b_{n-1})$. The proposition follows from the following lemma. ∎