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2014-05-18T21:23:02Z

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	~~In [[mathematics]], the '''theta divisor''' Θ~~ is ~~the [[divisor (algebraic geometry)\|divisor]] in the sense of [[algebraic geometry]] defined on an [[abelian variety]] ''A'' over the complex numbers (~~and ~~[[principally polarized]]) by the zero locus of the associated [[Riemann theta-function]]~~. It is ~~therefore an~~ [~~[algebraic subvariety]] of ''A'' of dimension dim ''A'' − 1~~.		Wilber Berryhill is what his wife enjoys to contact him and he completely loves this title. Kentucky is exactly [http://formalarmour.com/index.php?do=/profile-26947/info/ free psychic reading] where I've always been residing. Credit authorising is exactly where my main income arrives from. It's not a typical factor but what I like performing is to climb but I don't have the time lately.<br><br>Here is [http://brazil.amor-amore.com/irboothe are psychics real] my blog post; online psychic readings - [http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow/ mouse click the following web page] -

	=~~=Classical theory==~~

	~~Classical results of [[Bernhard Riemann~~]~~] describe Θ in another way, in the case that ''A'~~' is ~~the [[Jacobian variety]] ''J'' of an [[algebraic curve]] ([[compact Riemann surface]]) ''C''~~. ~~There is, for a choice of base point~~ '~~'P'' on ''C'',~~ a ~~standard mapping of ''C''~~ to '~~'J'', by means of~~ the ~~interpretation of ''J'' as the [[linear equivalence]] classes of divisors on ''C'' of degree 0~~. ~~That is, ''Q'' on ''C'' maps to the class of ''Q'' − ''P''. Then since ''J'' is an [[algebraic group]], ''C'' may be added to itself ''k'' times on ''J'', giving rise to subvarieties ''W''~~<~~sub~~>~~''k''~~<~~/sub~~>.

	~~If ''g''~~ is ~~the~~ [[genus (mathematics)\|genus]] of ''C'', Riemann proved that Θ is a translate on ''J'' of ''W''<sub>''g'' − 1</sub>. He also described which points on ''W''<sub>''g'' − 1</sub> are [[non-singular]]: they correspond to the effective divisors ''D'' of degree ''g'' − 1 with no associated meromorphic functions other than constants. In more classical language, these ''D'' do not move in a [[linear system of divisors]] on ''C'', in the sense that they do not dominate the polar divisor of a non constant function.

	~~Riemann further proved the '''Riemann singularity theorem''', identifying the [[multiplicity of a point]] p = class(D) on ''W''<sub>''g'' − 1<~~/~~sub> as the number of independent meromorphic functions with pole divisor dominated by D, or equivalently as ''h''<sup>0<~~/~~sup>(O(D)), the number of independent [[global section]]s of the [[holomorphic line bundle]] associated to ''D'' as [[Cartier divisor]] on ''C''~~.

	~~==Later work==~~

	~~The Riemann singularity theorem was extended by [[George Kempf]] in 1973,<ref>{{cite journal \| author=G. Kempf \| title=On the geometry of a theorem of Riemann \| journal=[[Ann. Of Math~~.~~]] \| volume=98 \| year=1973 \| pages=178–185 \| doi=10.2307~~/~~1970910 \| jstor=1970910 \| issue=1}}</ref> building on work of [[David Mumford]~~] ~~and Andreotti - Mayer, to a description of the singularities of points p = class(D) on ''W''<sub>''k''</sub> for 1 ≤ ''k'' ≤ ''g'' &minus~~; ~~1. In particular he computed their multiplicities also in terms of the number of independent meromorphic functions associated to D ('''Riemann~~-~~Kempf singularity theorem''').<ref>Griffiths and Harris, p.348</ref>~~

	~~More precisely, Kempf mapped ''J'' locally near ''p'' to a family of matrices coming from an~~ [~~[exact sequence]] which computes ''h''<sup>0<~~/~~sup>(O(D)), in such a way that ''W''<sub>''k''<~~/~~sub> corresponds to the locus of matrices of less than maximal rank. The multiplicity then agrees with that of the point on the corresponding rank locus~~. ~~Explicitly, if~~

	~~:''h''<sup>0<~~/~~sup>(O(D)) = ''r'' + 1,~~

	~~the multiplicity of ''W''<sub>''k''<~~/~~sub> at class(D) is the binomial coefficient~~

	~~:<math>{g~~-~~k+r \choose r}.</math>~~

	~~When ''d'' = ''g'' − 1, this is ''r'' + 1, Riemann's formula.~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	* {{cite book \| author=P. Griffiths \| authorlink=Phillip Griffiths \| coauthors=[[Joe Harris (mathematician)\|J. Harris]] \| title=Principles of Algebraic Geometry \| series=Wiley Classics Library \| publisher=Wiley Interscience \| year=1994 \| isbn=0-~~471~~-~~05059~~-~~8 }}~~

	~~[[Category:Theta functions]]~~
	~~[[Category:Algebraic curves]~~]

en>Qetuth: more specific stub type

2012-01-01T01:50:41Z

more specific stub type

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In [[mathematics]], the '''theta divisor''' Θ is the [[divisor (algebraic geometry)|divisor]] in the sense of [[algebraic geometry]] defined on an [[abelian variety]] ''A'' over the complex numbers (and [[principally polarized]]) by the zero locus of the associated [[Riemann theta-function]]. It is therefore an [[algebraic subvariety]] of ''A'' of dimension dim ''A'' − 1.

==Classical theory==

Classical results of [[Bernhard Riemann]] describe Θ in another way, in the case that ''A'' is the [[Jacobian variety]] ''J'' of an [[algebraic curve]] ([[compact Riemann surface]]) ''C''. There is, for a choice of base point ''P'' on ''C'', a standard mapping of ''C'' to ''J'', by means of the interpretation of ''J'' as the [[linear equivalence]] classes of divisors on ''C'' of degree 0. That is, ''Q'' on ''C'' maps to the class of ''Q'' − ''P''. Then since ''J'' is an [[algebraic group]], ''C'' may be added to itself ''k'' times on ''J'', giving rise to subvarieties ''W''''k''.

If ''g'' is the [[genus (mathematics)|genus]] of ''C'', Riemann proved that Θ is a translate on ''J'' of ''W''''g'' − 1. He also described which points on ''W''''g'' − 1 are [[non-singular]]: they correspond to the effective divisors ''D'' of degree ''g'' − 1 with no associated meromorphic functions other than constants. In more classical language, these ''D'' do not move in a [[linear system of divisors]] on ''C'', in the sense that they do not dominate the polar divisor of a non constant function.

Riemann further proved the '''Riemann singularity theorem''', identifying the [[multiplicity of a point]] p = class(D) on ''W''''g'' − 1 as the number of independent meromorphic functions with pole divisor dominated by D, or equivalently as ''h''0(O(D)), the number of independent [[global section]]s of the [[holomorphic line bundle]] associated to ''D'' as [[Cartier divisor]] on ''C''.

==Later work==

The Riemann singularity theorem was extended by [[George Kempf]] in 1973,<ref>{{cite journal | author=G. Kempf | title=On the geometry of a theorem of Riemann | journal=[[Ann. Of Math.]] | volume=98 | year=1973 | pages=178–185 | doi=10.2307/1970910 | jstor=1970910 | issue=1}}</ref> building on work of [[David Mumford]] and Andreotti - Mayer, to a description of the singularities of points p = class(D) on ''W''''k'' for 1 ≤ ''k'' ≤ ''g'' − 1. In particular he computed their multiplicities also in terms of the number of independent meromorphic functions associated to D ('''Riemann-Kempf singularity theorem''').<ref>Griffiths and Harris, p.348</ref>

More precisely, Kempf mapped ''J'' locally near ''p'' to a family of matrices coming from an [[exact sequence]] which computes ''h''0(O(D)), in such a way that ''W''''k'' corresponds to the locus of matrices of less than maximal rank. The multiplicity then agrees with that of the point on the corresponding rank locus. Explicitly, if

:''h''0(O(D)) = ''r'' + 1,

the multiplicity of ''W''''k'' at class(D) is the binomial coefficient

:<math>{g-k+r \choose r}.</math>

When ''d'' = ''g'' − 1, this is ''r'' + 1, Riemann's formula.

==Notes==
{{reflist}}

==References==
* {{cite book | author=P. Griffiths | authorlink=Phillip Griffiths | coauthors=[[Joe Harris (mathematician)|J. Harris]] | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 }}

[[Category:Theta functions]]
[[Category:Algebraic curves]]

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en>Crasshopper at 21:23, 18 May 2014

en>Qetuth: more specific stub type