Suppose we have a compact Riemann surface $X$ and a positive divisor $D > 0$ on $X$. Suppose further that $\dim L(D) = 1 + \deg(D)$ and that we have a point $p \in X$ such that $\dim L(p) = 2$. It is possible to conclude that $X\cong \mathbb{C}_{\infty}$, where $\mathbb{C}_{\infty}$ is the Riemann sphere?
Isomorphism to Riemann sphere
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complex-analysis
algebraic-geometry
riemann-surfaces
1 Answers
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If $\dim L(p)=2$, you know that there is a non constant map in $L(p)$. Let $f$ be such a map, because it is non constant, $f$ has at least one pole. On the other hand, we know that $f\in L(p)$, so it has at most one pole (at $p$) and that this pole is of order at most 1. So $f$ has a single pole which is of order 1. Thus, it is a map $X\rightarrow\mathbb{C}_\infty$ of degree 1, and so is an isomorphism.