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]).

Comments

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].

References

[1]	A. Andreotti, A. Mayer, "On period relations for abelian integrals on algebraic curves" Ann. Sci. Scuola Norm. Sup. Pisa , 21 : 2 (1967) pp. 189–238
[a1]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978)
[a2]	D. Mumford, "Curves and their Jacobians" , Univ. Michigan Press (1975)
[a3]	E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , 1 , Springer (1985)
[a4]	E. Arbarello, "Fay's triscant formula and a characterisation of Jacobian varieties" S.J. Bloch (ed.) , Algebraic geometry (Bowdoin, 1985) , Proc. Symp. Pure Math. , 46 , Amer. Math. Soc. (1987) pp. 49–61
[a5]	R.C. Gunning, "On theta functions for Jacobi varieties" S.J. Bloch (ed.) , Algebraic geometry (Bowdoin, 1985) , Proc. Symp. Pure Math. , 46 , Amer. Math. Soc. (1987) pp. 89–98