I am reading example IV.5.5.3 from Hartshorne and there's a line that I don't understand: it says that suppose we have a curve $X$ a genus 5 in its canonical embedding as a cuve of degree 8 in $\mathbb{P}^4$, and suppose $P,Q,R \in X$, then since $X$ is in its canonical embedding, $\operatorname{dim}(|K-P-Q-R|)$ is the dimenison of the linear system of hyperplanes in $\mathbb{P}^4$ which contain $P,Q,R$.
Why it this true? I guess I am not sure how to interpret the dimension of $|K-P-Q-R|$ geometrically.
The linear system $|K−P−Q−R|$ corresponds to divisors of sections of the line bundle $\mathcal{O}_X(K)$ that vanish on $P,Q,R$.
Let $(y_0:\dots:y_4)$ be homogeneous coordinates of $\mathbb{P}^4$. Since $X$ is embedded by $$ x \in X \mapsto (s_0(x): \dots: s_4(x)) \in \mathbb{P}^4 $$ using a basis $\{s_0, \dots, s_4 \}$ of ${\rm H}^0(X, \mathcal{O}_X(K))$, any section $s = a_0s_0 + \cdots + a_4s_4$ that vanishes on a point $p\in X$ corresponds to the hyperplane $\{ a_0y_0 + \cdots + a_4y_4= 0\}$ that contains $(s_0(p): \dots: s_4(p))$.