# Poincaré divisor

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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 $\mathop{\rm dim} _ {\mathbf C} \sing \Theta \geq g - 4$, where $g$ is the genus of the curve $X$( see [1]).

#### 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