View source for Poincaré divisor

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073010/p0730101.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073010/p0730102.png" /> of the curve. Therefore the effective divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073010/p0730103.png" /> given by this polarization is uniquely determined up to translation by an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073010/p0730104.png" />. The geometry of the Poincaré divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073010/p0730105.png" /> reflects the geometry of the algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073010/p0730106.png" />. In particular, the set of singular points of the Poincaré divisor has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073010/p0730107.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073010/p0730108.png" /> is the genus of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073010/p0730109.png" /> (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>