Computing $l(D)$ for certain divisor.

Question

Let $C$ be a smooth projective curve of genus $g=2$. I want to prove that there exist $P,Q\in C$ such that $$ l(P+Q)=2. $$ I know that if $D\in Div(C)$, and $x\in C$, then $$ l(D)\leq l(D+x)\leq l(D)+1. $$ According to this, if we consider $D=0$, we have that $$ 1\leq l(P+Q)\leq 3. $$ How could we follow from here?

Answer 1

Let $K$ be a canonical divisor. We have $\operatorname{deg} K=2g-2=2$ and $l(K)=2$. In particular it is linearly equivalent to an effective divisor $E$. Because $E$ is effective of degree 2, it is of the form $E=P+Q$ where $P$ and $Q$ are two not necessarily distinct points.

We will show that we can take $P$ and $Q$ distinct. So assume that $P=Q$.

Take any non constant section of $L(E)$, it can be seen of a rational map $f:C\rightarrow\mathbb{P}^1$ with poles at $P$ only and of order at most 2 by assumption. Note that $f$ has a pole since it is not constant, and it is of order 2 otherwise $f$ would be of degree 1, which contradict the fact that $C$ is not rational.

The map $f$ is not ramified everywhere (in characteristic $\neq 2$), this means that there exist $a\in k$ such that the fiber of $f$ at $a$ is two distinct points $A$ and $B$ (if $k$ is algebraically closed). Consider $f-a$ : $\operatorname{div}(f-a)=A+B-2P$, so that $2P$ is equivalent to $A+B$ and $l(A+B)=l(2P)=2$.

Answer 2

As Roland says, $deg\ K=2g-2=2$. Also, by Riemann Roch, $h^0(C,K)-h^0(C,\mathcal{O})=2+1-2=1$. Hence $h^0(C,K)=2$, so $dim|K|=1$. Hence we can find an effective $D$ linearly equivalent to $K$.