Describe a map $\rho : C^{(3)}\longrightarrow \operatorname{Pic}^3(C)$

Question

Let $C$ be a smooth curve of genus 3. I want to describe a map $\rho : C^{(3)}\longrightarrow \operatorname{Pic}^3(C)$, which maps every three points to a divisor of degree 3. Rhiemann-Roch theorem provides that

$$\operatorname{dim}\lvert D \rvert = \operatorname{dim}\lvert K_C-D \rvert + 1$$

Then for $D=K_C-P$, where $P$ is a point on curve, we have $\operatorname{dim}\lvert K_C-P \rvert=1$. I know, that for $D\neq K_C-P$ dimension of a linear system $\lvert K_C-P \rvert$ should be 0, but cannot show this.

P.S. original task from Birkenhake C., Lange H. - Complex abelian varieties

Answer 1

First, typically one uses the notation $|D|$ for $h^0(D)-1$.

If $h^0(D)>1$, RR implies $h^0(K-D)>0$. So, $K-D$ is effective, but it has degree one and so $K-D=P$ for some point $P$. So, $D=K-P$.