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Difference between revisions of "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 <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]]]).
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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) $
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given by this polarization is uniquely determined up to translation by an element $  x \in J ( X) $.  
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The geometry of the Poincaré divisor $  \Theta $
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reflects the geometry of the algebraic curve $  X $.  
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In particular, the set of singular points of the Poincaré divisor has dimension $  \mathop{\rm dim} _ {\mathbf C}  \sing  \Theta \geq  g - 4 $,  
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where $  g $
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is the genus of the curve $  X $(
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see [[#References|[1]]]).
  
 
====References====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 08:06, 6 June 2020


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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_divisor&oldid=48202
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article