Let $X$ be a smooth projective complex variety. Assume that $Z$ is a subvariety. Let $T$ be a generic complete intersection of codimension $\dim Z-1$. Assume that $p$ is a point in $Z_T:=T\cap Z$. Is there a formula relating $mult_p(Z)$ and $mult_p(Z_T)$?
$\begingroup$
$\endgroup$
5
-
2$\begingroup$ I don't understand the role of $X$. In order for "complete intersection" to make sense, you need to embed everything in $\mathbb{P}^n$. So why not take $X=\mathbb{P}^n$? $\endgroup$– Laurent Moret-BaillyCommented Apr 28, 2011 at 7:20
-
$\begingroup$ You can take $X=\mathbb{P}^n$. For general $X$, you can think the complete intersection as the intersection of general hyperplane sections. $\endgroup$– Fei YECommented Apr 28, 2011 at 22:07
-
$\begingroup$ If $P$ is generic enough, then $mult_P(Z) = mult_P(Z_T) = 1$. $\endgroup$– pinakiCommented Jul 18, 2011 at 12:10
-
$\begingroup$ Dear Auniket, Thank you very much. Can you give me some hints on how to prove it? $\endgroup$– Fei YECommented Jul 19, 2011 at 9:19
-
$\begingroup$ I am sorry to reply so late. If by multiplicity you mean what is defined e.g. in here: eom.springer.de/M/m065500.htm (or chapter 5 of Mumford's Algebraic Geometry I), then a point on a variety has multiplicity one iff it is non-singular. My claim in the preceding comment follows from this. But then it is true for ALL subvarieties of Z (i.e. not necessarily complete intersections). Therefore I suspect you used some other notions of multiplicity, e.g. the one in Qing Liu's answer. $\endgroup$– pinakiCommented Jul 26, 2011 at 0:33
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
The answer is yes (auniket's comment meant probably $T$ is generic and not $p$). This is a classical result : if $A$ is a noetherian local ring of dimension $d>0$, with infinite residue field, then there exists a system of parameters $(f_1,...,f_d)$ such that $\mathrm{mult}(A)=\mathrm{mult}(A/(f_1,...,f_{d-1}))$, see Zariski-Samuel vol. II, Chap. VIII, §10, Theorem 22 and Remark page 296.
Note that the result is false if the residue field is finite. But it is true that $\mathrm{mult}_p(Z)$ is a finite integral combination (with possibly negative coefficients) of multiplicities at $p$ of $Z\cap T_i$'s. See Proposition 5.9 in here.