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Poincaré divisor

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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 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 of the curve. Therefore the effective divisor given by this polarization is uniquely determined up to translation by an element . The geometry of the Poincaré divisor reflects the geometry of the algebraic curve . In particular, the set of singular points of the Poincaré divisor has dimension , where is the genus of the curve (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:
Poincare divisor. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Poincare_divisor&oldid=23469