Embed $\mathbb{P}^1$ with $n$ marked points in $\mathbb{P}^{n-2}$

Question

I read (see Inv. to Quantum Coh. by Vainsencher, p. 40) that

A $\mathbb{P}^1$ with $n$ marked points $P_1,\ldots,P_n$ is embedded in $\mathbb{P}^{n-2}$ by the linear system $|K_{\mathbb{P}^1} + P_1 + \ldots + P_n| = |\mathcal{O}(n-2)|$.

What is the embedding, explicitly?

Answer 1

$$t \mapsto \bigg[ \frac{p_n - p_1}{t - p_1} : \frac{p_n - p_2}{t - p_2} : \cdots : \frac{p_n - p_{n-1}}{t - p_{n-1}} \bigg]$$