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Difference between revisions of "Point in general position"

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A point on an algebraic variety that belongs to an open and dense subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073170/p0731701.png" /> in the Zariski topology. In algebraic geometry a point in general position is often called simply a generic point.
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A point on an algebraic variety that belongs to an open and dense subset $S$ in the Zariski topology. In algebraic geometry a point in general position is often called simply a generic point.
  
  
  
 
====Comments====
 
====Comments====
More precisely, a point is said to be in general position if it is outside a certain (given, or to be described) closed set; a point in an irreducible set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073170/p0731702.png" /> is called generic if it is outside every closed set different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073170/p0731703.png" /> itself.
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More precisely, a point is said to be in general position if it is outside a certain (given, or to be described) closed set; a point in an irreducible set $X$ is called generic if it is outside every closed set different from $X$ itself.
  
In differential topology, the phrase "a point in general position" often is used in the sense of a generic point, which is roughly "a point with no particular relationship of current importance to other structure elements being considered" . The precise meaning depends on the context. An element generically has a certain property if the property holds outside (a countable intersection of) open dense set(s). For instance, a polynomial has generically no double roots. Two submanifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073170/p0731704.png" /> are in general position if they "intersect as little as possible" . If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073170/p0731705.png" />, this means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073170/p0731706.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073170/p0731707.png" />, then for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073170/p0731708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073170/p0731709.png" />. Any non-general position situation can be changed to a general position situation by an arbitrarily small change, while if things are in general position, then sufficiently small changes do not change that. A precise version of general position is [[Transversality|transversality]], cf. also [[Transversality condition|Transversality condition]].
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In differential topology, the phrase "a point in general position" often is used in the sense of a generic point, which is roughly "a point with no particular relationship of current importance to other structure elements being considered" . The precise meaning depends on the context. An element generically has a certain property if the property holds outside (a countable intersection of) open dense set(s). For instance, a polynomial has generically no double roots. Two submanifolds $A,B\subset N$ are in general position if they "intersect as little as possible" . If $\dim A+\dim B<\dim N$, this means $A\cap B=\emptyset$; if $\dim A+\dim B\geq\dim N$, then for every $x\in A\cap B$, $T_xA+T_xB=T_xN$. Any non-general position situation can be changed to a general position situation by an arbitrarily small change, while if things are in general position, then sufficiently small changes do not change that. A precise version of general position is [[Transversality|transversality]], cf. also [[Transversality condition|Transversality condition]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977) pp. 91</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M.W. Hirsch,   "Differential topology" , Springer (1976) pp. 4, 78</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D.R.J. Chillingworth,   "Differential topology with a view to applications" , Pitman (1976) pp. 221ff</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M.W. Hirsch, "Differential topology" , Springer (1976) pp. 4, 78 {{MR|0448362}} {{ZBL|0356.57001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D.R.J. Chillingworth, "Differential topology with a view to applications" , Pitman (1976) pp. 221ff {{MR|0646088}} {{ZBL|0336.58001}} </TD></TR></table>

Latest revision as of 12:46, 4 September 2014

A point on an algebraic variety that belongs to an open and dense subset $S$ in the Zariski topology. In algebraic geometry a point in general position is often called simply a generic point.


Comments

More precisely, a point is said to be in general position if it is outside a certain (given, or to be described) closed set; a point in an irreducible set $X$ is called generic if it is outside every closed set different from $X$ itself.

In differential topology, the phrase "a point in general position" often is used in the sense of a generic point, which is roughly "a point with no particular relationship of current importance to other structure elements being considered" . The precise meaning depends on the context. An element generically has a certain property if the property holds outside (a countable intersection of) open dense set(s). For instance, a polynomial has generically no double roots. Two submanifolds $A,B\subset N$ are in general position if they "intersect as little as possible" . If $\dim A+\dim B<\dim N$, this means $A\cap B=\emptyset$; if $\dim A+\dim B\geq\dim N$, then for every $x\in A\cap B$, $T_xA+T_xB=T_xN$. Any non-general position situation can be changed to a general position situation by an arbitrarily small change, while if things are in general position, then sufficiently small changes do not change that. A precise version of general position is transversality, cf. also Transversality condition.

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 MR0463157 Zbl 0367.14001
[a2] M.W. Hirsch, "Differential topology" , Springer (1976) pp. 4, 78 MR0448362 Zbl 0356.57001
[a3] D.R.J. Chillingworth, "Differential topology with a view to applications" , Pitman (1976) pp. 221ff MR0646088 Zbl 0336.58001
How to Cite This Entry:
Point in general position. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Point_in_general_position&oldid=17545
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article