https://en.formulasearchengine.com/index.php?title=Coframe&feed=atom&action=history
Coframe - Revision history
2024-03-28T19:30:34Z
Revision history for this page on the wiki
MediaWiki 1.42.0-wmf.5
https://en.formulasearchengine.com/index.php?title=Coframe&diff=244597&oldid=prev
en>Crasshopper at 21:23, 18 May 2014
2014-05-18T21:23:02Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 22:23, 18 May 2014</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In [[mathematics]], the '''theta divisor''' Θ </del>is <del style="font-weight: bold; text-decoration: none;">the [[divisor (algebraic geometry)|divisor]] in the sense of [[algebraic geometry]] defined on an [[abelian variety]] ''A'' over the complex numbers (</del>and <del style="font-weight: bold; text-decoration: none;">[[principally polarized]]) by the zero locus of the associated [[Riemann theta-function]]</del>. <del style="font-weight: bold; text-decoration: none;">It </del>is <del style="font-weight: bold; text-decoration: none;">therefore an </del>[<del style="font-weight: bold; text-decoration: none;">[algebraic subvariety]] of ''A'' of dimension dim ''A'' &minus; 1</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Wilber Berryhill </ins>is <ins style="font-weight: bold; text-decoration: none;">what his wife enjoys to contact him </ins>and <ins style="font-weight: bold; text-decoration: none;">he completely loves this title</ins>. <ins style="font-weight: bold; text-decoration: none;">Kentucky </ins>is <ins style="font-weight: bold; text-decoration: none;">exactly </ins>[<ins style="font-weight: bold; text-decoration: none;">http://formalarmour.com/index</ins>.<ins style="font-weight: bold; text-decoration: none;">php?do</ins>=<ins style="font-weight: bold; text-decoration: none;">/profile-26947/info/ free psychic reading</ins>] <ins style="font-weight: bold; text-decoration: none;">where I</ins>'<ins style="font-weight: bold; text-decoration: none;">ve always been residing. Credit authorising </ins>is <ins style="font-weight: bold; text-decoration: none;">exactly where my main income arrives from</ins>. <ins style="font-weight: bold; text-decoration: none;">It</ins>'<ins style="font-weight: bold; text-decoration: none;">s not </ins>a <ins style="font-weight: bold; text-decoration: none;">typical factor but what I like performing is </ins>to <ins style="font-weight: bold; text-decoration: none;">climb but I don</ins>'<ins style="font-weight: bold; text-decoration: none;">t have </ins>the <ins style="font-weight: bold; text-decoration: none;">time lately</ins>.<<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">Here </ins>is <ins style="font-weight: bold; text-decoration: none;"> </ins>[<ins style="font-weight: bold; text-decoration: none;">http</ins>://<ins style="font-weight: bold; text-decoration: none;">brazil</ins>.<ins style="font-weight: bold; text-decoration: none;">amor-amore</ins>.<ins style="font-weight: bold; text-decoration: none;">com</ins>/<ins style="font-weight: bold; text-decoration: none;">irboothe are psychics real</ins>] <ins style="font-weight: bold; text-decoration: none;">my blog post</ins>; <ins style="font-weight: bold; text-decoration: none;">online psychic readings </ins>- [<ins style="font-weight: bold; text-decoration: none;">http:</ins>//<ins style="font-weight: bold; text-decoration: none;">myoceancounty</ins>.<ins style="font-weight: bold; text-decoration: none;">net</ins>/<ins style="font-weight: bold; text-decoration: none;">groups</ins>/<ins style="font-weight: bold; text-decoration: none;">apply-these-guidelines</ins>-<ins style="font-weight: bold; text-decoration: none;">when</ins>-<ins style="font-weight: bold; text-decoration: none;">gardening</ins>-<ins style="font-weight: bold; text-decoration: none;">and</ins>-<ins style="font-weight: bold; text-decoration: none;">grow/ mouse click the following web page</ins>] <ins style="font-weight: bold; text-decoration: none;">-</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>=<del style="font-weight: bold; text-decoration: none;">=Classical theory==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Classical results of [[Bernhard Riemann</del>]<del style="font-weight: bold; text-decoration: none;">] describe Θ in another way, in the case that ''A'</del>' is <del style="font-weight: bold; text-decoration: none;">the [[Jacobian variety]] ''J'' of an [[algebraic curve]] ([[compact Riemann surface]]) ''C''</del>. <del style="font-weight: bold; text-decoration: none;">There is, for a choice of base point </del>'<del style="font-weight: bold; text-decoration: none;">'P'' on ''C'', </del>a <del style="font-weight: bold; text-decoration: none;">standard mapping of ''C'' </del>to '<del style="font-weight: bold; text-decoration: none;">'J'', by means of </del>the <del style="font-weight: bold; text-decoration: none;">interpretation of ''J'' as the [[linear equivalence]] classes of divisors on ''C'' of degree 0</del>. <del style="font-weight: bold; text-decoration: none;">That is, ''Q'' on ''C'' maps to the class of ''Q'' &minus; ''P''. Then since ''J'' is an [[algebraic group]], ''C'' may be added to itself ''k'' times on ''J'', giving rise to subvarieties ''W''</del><<del style="font-weight: bold; text-decoration: none;">sub</del>><del style="font-weight: bold; text-decoration: none;">''k''</del><<del style="font-weight: bold; text-decoration: none;">/sub</del>><del style="font-weight: bold; text-decoration: none;">.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">If ''g'' </del>is <del style="font-weight: bold; text-decoration: none;">the </del>[<del style="font-weight: bold; text-decoration: none;">[genus (mathematics)|genus]] of ''C'', Riemann proved that Θ is a translate on ''J'' of ''W''<sub>''g'' &minus; 1</sub>. He also described which points on ''W''<sub>''g'' &minus; 1</sub> are [[non-singular]]</del>: <del style="font-weight: bold; text-decoration: none;">they correspond to the effective divisors ''D'' of degree ''g'' &minus; 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. </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Riemann further proved the '''Riemann singularity theorem''', identifying the [[multiplicity of a point]] p = class(D) on ''W''<sub>''g'' &minus; 1<</del>/<del style="font-weight: bold; text-decoration: none;">sub> as the number of independent meromorphic functions with pole divisor dominated by D, or equivalently as ''h''<sup>0<</del>/<del style="font-weight: bold; text-decoration: none;">sup>(O(D)), the number of independent [[global section]]s of the [[holomorphic line bundle]] associated to ''D'' as [[Cartier divisor]] on ''C''</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Later work==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">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</del>.<del style="font-weight: bold; text-decoration: none;">]] | volume=98 | year=1973 | pages=178–185 | doi=10.2307</del>/<del style="font-weight: bold; text-decoration: none;">1970910 | jstor=1970910 | issue=1}}</ref> building on work of [[David Mumford]</del>] <del style="font-weight: bold; text-decoration: none;">and Andreotti - Mayer, to a description of the singularities of points p = class(D) on ''W''<sub>''k''</sub> for 1 ≤ ''k'' ≤ ''g'' &minus</del>; <del style="font-weight: bold; text-decoration: none;">1. In particular he computed their multiplicities also in terms of the number of independent meromorphic functions associated to D ('''Riemann</del>-<del style="font-weight: bold; text-decoration: none;">Kempf singularity theorem''').<ref>Griffiths and Harris, p.348</ref> </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">More precisely, Kempf mapped ''J'' locally near ''p'' to a family of matrices coming from an </del>[<del style="font-weight: bold; text-decoration: none;">[exact sequence]] which computes ''h''<sup>0<</del>/<del style="font-weight: bold; text-decoration: none;">sup>(O(D)), in such a way that ''W''<sub>''k''<</del>/<del style="font-weight: bold; text-decoration: none;">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</del>. <del style="font-weight: bold; text-decoration: none;"> Explicitly, if </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:''h''<sup>0<</del>/<del style="font-weight: bold; text-decoration: none;">sup>(O(D)) = ''r'' + 1,</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">the multiplicity of ''W''<sub>''k''<</del>/<del style="font-weight: bold; text-decoration: none;">sub> at class(D) is the binomial coefficient </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:<math>{g</del>-<del style="font-weight: bold; text-decoration: none;">k+r \choose r}.</math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">When ''d'' = ''g'' &minus; 1, this is ''r'' + 1, Riemann's formula.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Notes==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{reflist}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==References==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* {{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</del>-<del style="font-weight: bold; text-decoration: none;">471</del>-<del style="font-weight: bold; text-decoration: none;">05059</del>-<del style="font-weight: bold; text-decoration: none;">8 }}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Theta functions]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Algebraic curves]</del>]</div></td><td colspan="2" class="diff-side-added"></td></tr>
</table>
en>Crasshopper
https://en.formulasearchengine.com/index.php?title=Coframe&diff=11711&oldid=prev
en>Qetuth: more specific stub type
2012-01-01T01:50:41Z
<p>more specific stub type</p>
<p><b>New page</b></p><div>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'' &minus; 1.<br />
<br />
==Classical theory==<br />
<br />
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'' &minus; ''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>.<br />
<br />
If ''g'' is the [[genus (mathematics)|genus]] of ''C'', Riemann proved that Θ is a translate on ''J'' of ''W''<sub>''g'' &minus; 1</sub>. He also described which points on ''W''<sub>''g'' &minus; 1</sub> are [[non-singular]]: they correspond to the effective divisors ''D'' of degree ''g'' &minus; 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. <br />
<br />
Riemann further proved the '''Riemann singularity theorem''', identifying the [[multiplicity of a point]] p = class(D) on ''W''<sub>''g'' &minus; 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''.<br />
<br />
==Later work==<br />
<br />
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> <br />
<br />
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 <br />
<br />
:''h''<sup>0</sup>(O(D)) = ''r'' + 1,<br />
<br />
the multiplicity of ''W''<sub>''k''</sub> at class(D) is the binomial coefficient <br />
<br />
:<math>{g-k+r \choose r}.</math><br />
<br />
When ''d'' = ''g'' &minus; 1, this is ''r'' + 1, Riemann's formula.<br />
<br />
==Notes==<br />
{{reflist}}<br />
<br />
==References==<br />
* {{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 }}<br />
<br />
[[Category:Theta functions]]<br />
[[Category:Algebraic curves]]</div>
en>Qetuth