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 of a variety at a point is defined to be the multiplicity of the maximal ideal in the local ring . The multiplicity of at coincides with the multiplicity of the tangent cone at the vertex, and also with the degree of the special fibre of a blow-up of at , where is considered to be immersed in the projective space (see [3]). One has if and only if is a non-singular (regular) point of . If is a hypersurface in a neighbourhood of (i.e. is given by a single equation in an affine space ), then is identical with the number such that , where is the maximal ideal in the local ring . The multiplicity does not change when is cut by a generic hypersurface through . If denotes the set of points such that , then 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.