# 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 $ \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 |

#### 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

[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 |

**How to Cite This Entry:**

Poincaré divisor.

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