View source for Multiplicity of a singular point

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''of an algebraic variety''

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 $
at  $  x $
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 $
of  $  X $
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 $
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====
<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====
For the multiplicity of the maximal ideal of a local ring, cf. [[Multiplicity of a module|Multiplicity of a module]].