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Let $A$ be a principally polarised abelian variety of dimension $g$ over a number field. A theta divisor is a divisor of degree 0 on $A$. In the case $A$ is the Jacobian of a curve $C$ of genus $g$ over $K$, it can be defined as follows. We let $W$ be the image of $C^{g-1}$ in $\mathrm{Pic}^{g-1}(C)$ under the canonical map $C^{g-1} \to \mathrm{Pic}^{g-1}(C)$. We let $D \in \mathrm{Pic}^{g-1}(C)$ be a divisor such that $2 \cdot D$ is the canonical divisor class. Then $W - D$ is a theta divisor on $A$. Theta divisors are not unique: there are $2^g$ of them, their differences are 2-torsion divisors.

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

Knowl status:
  • Review status: beta
  • Last edited by Raymond van Bommel on 2019-12-13 10:35:39
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