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

Comments

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