Divisor on curve of genus 2

Question

Let $C$ be a smooth, projective curve of genus 2. I want to show that there exists a non-constant rational function $f \in k(C)$ having divisor of the form $$(f) = P_1 + P_2 - P_3 - P_4 $$for points $P_1,P_2,P_3,P_4$. I am given the hint to consider two rational functions $f_1, f_2 \in L(K)$ (the Riemann-Roch space) where $K$ is a canonical divisor on the curve. I know that $\dim L(K) = 2$, so these non-constant $f_1,f_2$ exist. Moreover, I can derive that $\deg(K)=2$. How do I use the latter to solve this? I thought about taking $K = P_3 + P_4$ as that would give me $f_1$ and $f_2$ both with simple poles at $P_3$ and $P_4$. Does this mean that I can take $f = \frac{f_1}{f_2}$ since it doesn't say that the $P_i$ should be different?

Answer 1

Take two linearly independent rational differential forms $\omega_1, \omega_2\in L(K)=\Omega^1(C)$ with divisors $\operatorname {div} (\omega_1)=P_1+P'_1$ and $\operatorname {div} (\omega_2)=P_2+P'_2$ .
The rational function $f=\frac {\omega_1}{\omega_2}\in \operatorname {Rat}(C)$ then has a divisor of the required form $$\operatorname {div}(f)=P_1+P'_1-P_2-P'_2 $$