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Difference between revisions of "Multiplicity of a singular point"

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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Mumford,   "Algebraic geometry" , '''1. Complex projective varieties''' , Springer (1976)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre,   "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer (1965)</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</TD></TR></table>
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<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>
  
  

Revision as of 21:54, 30 March 2012

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.

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=11768
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article