Poincaré divisor

The divisor given by the natural polarization over the Jacobian (cf. Jacobi variety) of an algebraic curve. The intersection form of one-dimensional cycles in the homology of an algebraic curve $X$ induces a unimodular skew-symmetric form on the lattice of periods. By the definition of a polarized Abelian variety (cf. Polarized algebraic variety) this form determines the principal polarization over the Jacobian $J(X)$ of the curve. Therefore the effective divisor $\Theta \subset J(X)$ given by this polarization is uniquely determined up to translation by an element $x \in J(X)$. The geometry of the Poincaré divisor $\Theta$ reflects the geometry of the algebraic curve $X$. In particular, the set of singular points of the Poincaré divisor has dimension $\operatorname{dim}_{\mathbf C} \operatorname{sing} \Theta \geq g-4$, where $g$ is the genus of the curve $X$ (see [1]).

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The above divisor is usually called the theta divisor of the Jacobi variety. For the rich geometry connected with it see, for instance, the books [a1], [a2] and [a3] and the survey articles [a4] and [a5].

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