View source for Poincaré divisor

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The [[Divisor|divisor]] given by the natural polarization over the Jacobian (cf. [[Jacobi variety|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|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 [[#References|[1]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR></table>

====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 [[#References|[a1]]], [[#References|[a2]]] and [[#References|[a3]]] and the survey articles [[#References|[a4]]] and [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , Wiley (Interscience)  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Mumford,   "Curves and their Jacobians" , Univ. Michigan Press  (1975)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Arbarello,   M. Cornalba,   P.A. Griffiths,   J.E. Harris,   "Geometry of algebraic curves" , '''1''' , Springer  (1985)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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</TD></TR></table>