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Let $C$ be a smooth projective curve over $\mathbb{C}$ and let $\Theta$ be the theta divisor in $J^{g-1}(C)$. The theta divisor is ample, so $J^{g-1}\setminus \Theta$ is affine. What is coordinate ring $R$ of this affine variety $$ J^{g-1}\setminus \Theta = \operatorname{Spec} R. $$

Is there a concrete description of $R$?

I'm mostly interested in the special case of a smooth plane quartic: $C \subset \mathbb{P}^2$, $\deg C =4$ and thus $g(C)=3$.

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