Let $C$ be a complex projective smooth hyperelliptic curve of genus $3$ and $A_1, A_2, A_3$ three distinct Weierstrass points on $C$. Consider the divisor $D=A_1+A_2+A_3$ and $L$ the line bundle associated to $D$. Question: is $h^0(C,L)$ bigger than one?
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$\begingroup$ To be more precise, $C$ is a Galois cover of degree two of a curve of genus 2. $\endgroup$– user95246Jan 9, 2020 at 16:42
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1 Answer
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If $A_1$, $A_2$ and $A_3$ are in distinct fibers of the double cover $C \to \mathbb P^1$, then $\mathrm h^0(C, L) = 1$. Otherwise you would get a map $C \to \mathbb P^1$ of degree 3; together with the double cover this would yield a map $C \to \mathbb P^1 \times \mathbb P^1$, whose image $X \subseteq \mathbb P^1 \times \mathbb P^1$ is a divisor of degree $(2,3)$, and such that the map $C \to X$ is birational. Since the arithmetic genus of $X$ is 2, this gives a contradiction.
