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''of an algebraic variety''
 
''of an algebraic variety''
  
An integer which is a measure of the singularity of the [[Algebraic variety|algebraic variety]] at that point. The multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m0655001.png" /> of a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m0655002.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m0655003.png" /> is defined to be the multiplicity of the maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m0655004.png" /> in the [[Local ring|local ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m0655005.png" />. The multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m0655006.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m0655007.png" /> coincides with the multiplicity of the [[Tangent cone|tangent cone]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m0655008.png" /> at the vertex, and also with the degree of the special fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m0655009.png" /> of a blow-up <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550011.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550013.png" /> is considered to be immersed in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550014.png" /> (see [[#References|[3]]]). One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550015.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550016.png" /> is a non-singular (regular) point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550017.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550018.png" /> is a hypersurface in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550019.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550020.png" /> is given by a single equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550021.png" /> in an affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550022.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550023.png" /> is identical with the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550026.png" /> is the maximal ideal in the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550027.png" />. The multiplicity does not change when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550028.png" /> is cut by a generic hypersurface through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550029.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550030.png" /> denotes the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065500/m06550033.png" /> is a closed subset (a subvariety).
+
An integer which is a measure of the singularity of the [[Algebraic variety|algebraic variety]] at that point. The multiplicity $  \mu ( X, x) $
 +
of a variety $  X $
 +
at a point $  x $
 +
is defined to be the multiplicity of the maximal ideal $  \mathfrak m $
 +
in the [[Local ring|local ring]] $  {\mathcal O} _ {X, x }  $.  
 +
The multiplicity of $  X $
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at $  x $
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coincides with the multiplicity of the [[Tangent cone|tangent cone]] $  C ( X, x) $
 +
at the vertex, and also with the degree of the special fibre $  \sigma  ^ {-} 1 ( x) $
 +
of a blow-up $  \sigma : X  ^  \prime  \rightarrow X $
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of $  X $
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at $  x $,  
 +
where $  \sigma  ^ {-} 1 ( X) $
 +
is considered to be immersed in the projective space $  P ( \mathfrak m / \mathfrak m ^ {2} ) $(
 +
see [[#References|[3]]]). One has $  \mu ( X, x) = 1 $
 +
if and only if $  x $
 +
is a non-singular (regular) point of $  X $.  
 +
If $  X $
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is a hypersurface in a neighbourhood of $  x $(
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i.e. $  X $
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is given by a single equation $  f = 0 $
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in an affine space $  Z $),  
 +
then $  \mu ( X, x) $
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is identical with the number $  n $
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such that $  f \in \mathfrak n  ^ {n} \setminus  \mathfrak n ^ {n + 1 } $,  
 +
where $  \mathfrak n $
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is the maximal ideal in the local ring $  {\mathcal O} _ {Z, x }  $.  
 +
The multiplicity does not change when $  X $
 +
is cut by a generic hypersurface through $  x $.  
 +
If $  X _ {d} $
 +
denotes the set of points $  x \in X $
 +
such that $  \mu ( X, x) \geq  d $,  
 +
then $  X _ {d} $
 +
is a closed subset (a subvariety).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Mumford, "Algebraic geometry" , '''1. Complex projective varieties''' , Springer (1976) {{MR|0453732}} {{ZBL|0356.14002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer (1965) {{MR|0201468}} {{ZBL|0142.28603}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> C.P. Ramanujam, "On a geometric interpretation of multiplicity" ''Invent. Math.'' , '''22''' : 1 (1973) pp. 63–67 {{MR|0354663}} {{ZBL|0265.14004}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Mumford, "Algebraic geometry" , '''1. Complex projective varieties''' , Springer (1976) {{MR|0453732}} {{ZBL|0356.14002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer (1965) {{MR|0201468}} {{ZBL|0142.28603}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> C.P. Ramanujam, "On a geometric interpretation of multiplicity" ''Invent. Math.'' , '''22''' : 1 (1973) pp. 63–67 {{MR|0354663}} {{ZBL|0265.14004}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
For the multiplicity of the maximal ideal of a local ring, cf. [[Multiplicity of a module|Multiplicity of a module]].
 
For the multiplicity of the maximal ideal of a local ring, cf. [[Multiplicity of a module|Multiplicity of a module]].

Revision as of 08:02, 6 June 2020


of an algebraic variety

An integer which is a measure of the singularity of the algebraic variety at that point. The multiplicity $ \mu ( X, x) $ of a variety $ X $ at a point $ x $ is defined to be the multiplicity of the maximal ideal $ \mathfrak m $ in the local ring $ {\mathcal O} _ {X, x } $. The multiplicity of $ X $ at $ x $ coincides with the multiplicity of the tangent cone $ C ( X, x) $ at the vertex, and also with the degree of the special fibre $ \sigma ^ {-} 1 ( x) $ of a blow-up $ \sigma : X ^ \prime \rightarrow X $ of $ X $ at $ x $, where $ \sigma ^ {-} 1 ( X) $ is considered to be immersed in the projective space $ P ( \mathfrak m / \mathfrak m ^ {2} ) $( see [3]). One has $ \mu ( X, x) = 1 $ if and only if $ x $ is a non-singular (regular) point of $ X $. If $ X $ is a hypersurface in a neighbourhood of $ x $( i.e. $ X $ is given by a single equation $ f = 0 $ in an affine space $ Z $), then $ \mu ( X, x) $ is identical with the number $ n $ such that $ f \in \mathfrak n ^ {n} \setminus \mathfrak n ^ {n + 1 } $, where $ \mathfrak n $ is the maximal ideal in the local ring $ {\mathcal O} _ {Z, x } $. The multiplicity does not change when $ X $ is cut by a generic hypersurface through $ x $. If $ X _ {d} $ denotes the set of points $ x \in X $ such that $ \mu ( X, x) \geq d $, then $ X _ {d} $ is a closed subset (a subvariety).

References

[1] D. Mumford, "Algebraic geometry" , 1. Complex projective varieties , Springer (1976) MR0453732 Zbl 0356.14002
[2] J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965) MR0201468 Zbl 0142.28603
[3] C.P. Ramanujam, "On a geometric interpretation of multiplicity" Invent. Math. , 22 : 1 (1973) pp. 63–67 MR0354663 Zbl 0265.14004

Comments

For the multiplicity of the maximal ideal of a local ring, cf. Multiplicity of a module.

How to Cite This Entry:
Multiplicity of a singular point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplicity_of_a_singular_point&oldid=47937
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article