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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.
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| ==Classical theory== | |
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| 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>.
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| 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.
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| 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''.
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| ==Later work==
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| 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'' − 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>
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| 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
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| :''h''<sup>0</sup>(O(D)) = ''r'' + 1,
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| the multiplicity of ''W''<sub>''k''</sub> at class(D) is the binomial coefficient
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| :<math>{g-k+r \choose r}.</math>
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| When ''d'' = ''g'' − 1, this is ''r'' + 1, Riemann's formula.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{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 }}
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| [[Category:Theta functions]]
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| [[Category:Algebraic curves]]
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