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Let $A$ be a principally polarized abelian variety of dimension $g$ over a field $k$. A theta divisor is an effective divisor $\Theta$ on $A_{\overline{k}}$ whose associated homomorphism $A_{\overline{k}} \to A^\vee_{\overline{k}}$ is the principal polarization. It is determined up to translation on $A_{\overline{k}}$. Any theta divisor is ample. The divisor $[-1]^* \Theta$ is another theta divisor, hence a translate of $\Theta$. If $\operatorname{char} k \ne 2$, then there are exactly $2^{2g}$ translates of $\Theta$ that are symmetric, meaning that $[-1]^* \Theta = \Theta$.

If $A$ is the Jacobian of a curve $C$ of genus $g$ over $k=\overline{k}$, then a theta divisor can be constructed as as the image of $C^{g-1} \to \mathrm{Pic}^{g-1}_C \stackrel{\sim}\to A$, where the last isomorphism is subtraction by the class of some fixed degree $g-1$ divisor.

A symmetric theta divisor on $A$ descends to a divisor on the Kummer variety $A / \{\pm 1\}$.

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• Review status: beta
• Last edited by Raymond van Bommel on 2022-06-29 21:59:38
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