en>Yobot: Tagging using AWB (10703)

2015-01-07T10:45:50Z

Tagging using AWB (10703)

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	~~In [[algebraic geometry]], an '''infinitely near point''' of an algebraic surface ''S'' is a point on a surface obtained from ''S'' by repeatedly blowing up points~~. ~~Infinitely near points of [[algebraic surface]]s were introduced~~ by ~~{{harvs\|txt\|first=Max\|last=Noether\|authorlink=Max Noether\|year=1876}}.<ref>''Infinitely Near Points on Algebraic Surfaces''~~, ~~Gino Turrin, ''American Journal of Mathematics'', Vol. 74, No. 1 (Jan., 1952), pp. 100–106</ref>~~		Hi there. Allow me start by introducing the author, her title is Sophia. Invoicing is my occupation. My wife and I reside in Mississippi but now I'm considering other choices. She is truly fond of caving but she doesn't have the time recently.<br><br>Here is my blog post real psychic readings - [http://jplusfn.gaplus.kr/xe/qna/78647 gaplus.kr] -

	~~There are some other meanings of "infinitely near point"~~. ~~Infinitely near points can also be defined for higher-dimensional varieties: there are several inequivalent ways to do this, depending on what one~~ is ~~allowed to blow up~~. Weil gave a definition of infinitely near points of smooth varieties,<ref>[4] Weil, A., ''Theorie des points proches sur les varietes differentielles'', Colloque de Topologie et Geometrie Diferentielle, Strasbourg, 1953, 111–117; in ~~his~~ ''Collected Papers'' II. The notes to the paper there indicate this was a rejected project for the [[Bourbaki group]]. Weil references [[Pierre de Fermat]]'s approach to calculus, as well as the jets of [[Charles Ehresmann]]. For an extended treatment, see O. O. Luciano, ''Categories of multiplicative functors and Weil's infinitely near points'', Nagoya Math. J. 109 (1988), 69–89 (online [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.nmj/1118780892 here] for a full discussion.</ref> though these are not the same as infinitely near points in algebraic geometry.
	~~In the line of [[hyperreal number]]s, an extension of the [[real number]] line, two points are called infinitely near if their difference~~ is ~~[[infinitesimal]].~~

	~~== Definition==~~

	~~When [[blowing up]] is applied to a point ''P'' on a surface ''S'', the new surface ''S''* contains a whole curve ''C'' where ''P'' used to be. The points~~ of '~~'C''~~ have the geometric interpretation as the tangent directions at ''P'' to ''S''. They can be called infinitely near to ''P'' as way of visualizing them on ''S'', rather than ''S''*. More generally this construction can be iterated by blowing up a point on the new curve ''C'', and so on.

	~~An '''infinitely near point''' (of order ''n'') ''P''~~<~~sub~~>~~''n''~~<~~/sub> on a surface ''S''<sub>0</sub~~> is given by a sequence of points ''P''<sub>0</sub>, ''P''<sub>1</sub>,...,''P''<sub>''n''</sub> on surfaces ''S''<sub>0</sub>, ''S''<sub>1</sub>,...,''S''<sub>''n''</sub> such that ''S''<sub>''i''</sub> is given by blowing up ''S''<sub>''i''–1</sub> at the point ''P''<sub>''i''–1</sub> and ''P''<sub>i</sub> is a point of the surface ''S''<sub>i</sub> with image ''P''<sub>''i''–1</sub>.

	~~In particular the points of the surface ''S'' are the infinitely near points on ''S'' of order 0.~~

	~~Infinitely near points correspond to 1~~-~~dimensional valuations of the function field of ''S'' with 0-dimensional center, and in particular correspond to some of the points of the~~ [[Zariski–Riemann surface]]. (The 1-dimensional valuations with 1-dimensional center correspond to irreducible curves of ''S''.) It is also possible to iterate the construction infinitely often, producing an infinite sequence ''P''<sub>0</sub>, ''P''<sub>1</sub>,... of infinitely near points. These infinite sequences correspond to the 0-dimensional valuations of the function field of the surface, which correspond to the "0-dimensional" points of the [[Zariski–Riemann surface]].

	~~==Applications==~~

	~~If ''C'' and ''D'' are distinct irreducible curves on a smooth surface ''S'' intersecting at a point ''p'', then the multiplicity of their intersection at ''p'' is given by~~
	:~~<math>\sum_{x \text{ infinitely near }p}m_x(C)m_x(D)<~~/~~math>~~
	~~where ''m''<sub>''x''<~~/~~sub>(''C'') is the multiplicity of ''C'' at ''x''~~. In general this is larger than ''m''<sub>''p''</sub>(''C'')''m''<sub>''p''</sub>(''D'') if ''C'' and ''D'' have a common tangent line at ''x'' so that they also intersect at infinitely near points of order greater than 0, for example if ''C'' is the line ''y''=0 and ''D'' is the parabola ''y''=''x''<sup>2</sup> and ''p''=(0,0).

	~~The genus of ''C'' is given by~~
	~~:<math> g(C)=g(N)+\sum_{\text{infinitely near points }x}m_x(m_x-1)/2<~~/~~math>~~
	~~where ''N'' is the normalization of ''C'' and ''m''<sub>''x''<~~/~~sub> is the multiplicity of the infinitely near point ''x'' on ''C''.~~

	~~==References==~~

	~~<references~~/>
	*{{citation\|first=M. ~~\|last=Noether\|title=Ueber die singularen Werthsysteme einer algebraischen Function und die singularen Punkte einer algebraischen Curve\|journal= Mathematische Annalen~~
	~~\|volume= 9 \|year=1876\|pages=166–182\|doi=10.1007/BF01443372}}~~

	~~[[Category:Geometry]]~~
	~~[[Category:Differential calculus]~~]
	~~[[Category:Non~~-~~standard analysis]]~~
	~~[[Category:Birational geometry]]~~

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en>Yobot: Tagging using AWB (10703)

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