# Multiplicity of a singular point

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