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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 $ X $
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The [[Divisor (algebraic geometry)|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|Polarized algebraic variety]]) this form determines the principal polarization over the Jacobian $ J ( 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) $
+
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) $.  
+
given by this polarization is uniquely determined up to translation by an element $x \in J(X)$.  
The geometry of the Poincaré divisor $ \Theta $
+
The geometry of the Poincaré divisor $\Theta$
reflects the geometry of the algebraic curve $ X $.  
+
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 $,  
+
In particular, the set of singular points of the Poincaré divisor has dimension $\operatorname{dim}_{\mathbf C}  \operatorname{sing} \Theta \geq g-4$, where $g$ is the genus of the curve $X$ (see [[#References|[1]]]).
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====
 
====Comments====
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====References====
 
====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>
+
<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>
 +
<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>

Latest revision as of 18:28, 17 April 2024


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 $\operatorname{dim}_{\mathbf C} \operatorname{sing} \Theta \geq g-4$, where $g$ is the genus of the curve $X$ (see [1]).

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

[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
[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