Exercise on Riemann surfaces

Question

Let $X$ be a compact Riemann surface of genus $g\geq 2$ and let $p,q\in X$ be two distinct points. I want to prove that:

$$\ell(p+q)\in \{1,2\}$$

Clearly $\ell(p+q)\geq 1$ because it contains holomorphic functions. Moreover by Riemann-Roch theorem:

$$g=\ell(K)\geq\ell(K-p-q)=g-3+\ell(p+q)$$

(I'll call the disequality above $\star$).

So I proved that:

$$\ell(p+q)\in \{1,2,3\}$$

I would just need to prove that the disequality $\star$ is actually strict to end this exercise. I'd need to prove that $$i(p+q)< i(0)=g$$ (i.e. that there are holomorphic forms that don't vanish at two fixed points on a Riemann surface). How can I do this?

Answer 1

As pointed out in the comments, since $g \geq 2$, then $\ell(p) = 1$: a nonconstant function $f \in L(p)$ would have a simple pole at $p$ and no other poles, hence would give an isomorphism $f: X \to \mathbb{P}^1$, contradiction. Thus $L(p)$ consists of only the constant functions. Since $L(p) \subseteq L(p+q)$, this gives the lower bound $$ 1 = \ell(p) \leq \ell(p+q) \, . $$

Now recall the result that, given any divisor $D$ and any point $x \in X$, we have $$ \ell(D+x) \leq \ell(D) + 1 \, $$ (See this post, this post, or Lemma V.3.15 (p. 151) in Miranda's Algebraic Curves and Riemann Surfaces for a proof.) Thus $$ \ell(p+q) \leq \ell(p) + 1 = 1 + 1 = 2 \, . $$

Answer 2

Indeed, it suffices to prove that for arbitrary distinct $p,q \in X$ there is a holomorphic form that does not vanish at both $p$ and $q$. However, it is simpler to prove the stronger fact: for arbitrary $p \in X$ there is a holomorphic form that does not vanish at $p$.

Indeed, assume that all holomorphic forms vanish at $p$. Then $i(p) = i(0) = g$, so $l(p) = 2$ by Riemann-Roch theorem. Thus, there exists a non-trivial meromorphic function that has exactly one simple pole at $p$. It is well-known that such a function does not exist on a compact Riemann surface if its genus is positive (i.e. it exists only on a sphere).

However, you could avoid using Riemann-Roch theorem at all by using the foregoing fact. Indeed, assume that there are two non-constant meromorphic functions $f$ and $g$ that have simple poles only at $p$ and $q$. It is easy to see that there is $c \in \mathbb C$ such that $f - cg$ does not have a pole at $p$. Then this function is a constant. Thus, $f = cg + d$ for some $c,d \in \mathbb C$. Therefore, there are at most two linear independent functions with only simple poles at $p$ and $q$.

Finally, note that $g = 1$ also works in your exercise.