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Let $C$ be a complex algebraic curve. It is well known that if $L$ is a special divisor on $C$, i.e., $h^0(L) > 0$ and $h^1(L) > 0$, then $$h^0 (L) \le \frac{1}{2} \deg L + 1.$$ Assume that $L$ is not trivial and $L \ne K_C$. The equality holds if and only if $C$ is hyperelliptic.

My question is that: if there is a nontrivial special divisor $L$ on $C$ such that $L \ne K_C$ $$h^0 (L) = \frac{1}{2} \deg L$$ holds, then what conditions $C$ should satisfy? For example, $C$ could be hyperelliptic. Is there a complete classification?

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# Classification of certain algebraic curves

Let $C$ be a complex algebraic curve. It is well known that if $L$ is a special divisor on $C$, i.e., $h^0(L) > 0$ and $h^1(L) > 0$, then $$h^0 (L) \le \frac{1}{2} \deg L + 1.$$ Assume that $L$ is not trivial. The equality holds if and only if $C$ is hyperelliptic.

My question is that: if $$h^0 (L) = \frac{1}{2} \deg L$$ holds, then what conditions $C$ should satisfy? For example, $C$ could be hyperelliptic. Is there a complete classification?