Let $X$ be a smooth plane curve of genus $3$ (assume a smooth plane quartic) and $D$ a divisor of degree $2$ on this curve.
Assume that $\mathcal{l}(D)>0$. It means that there exists a rational function $f$ such that $div(f)+D\geqslant 0$, hence $D$ is linearly equivalent to $P+Q$ (since it is of degree $2$). Why it implies that $\mathcal{l}(D)=1$?
Another question: assume that $D$ is an odd theta characteristic (it means $2D\simeq K_X$ where $K_X$ is a canonical divisor). From the previous question it would follow that $D$ is of equivalent to $P+Q$. Why then the line passing through $P$ and $Q$ is bitangent to $X$?