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Suppose $X$, $Y$, $Z$ are projective varieties in $\mathbb{P}^n_K$ of dimension $n-1$, where $K$ is a field. $X$, $Y$, $Z$ intersect properly, and $P$ is one of their intersection irreducible components of dimension $n-3$. Suppose their intersection multiplicity at $P$ is $i(P; X,Y,Z; \mathbb{P}^n)$. Do we always have $i(P; X,Y,Z; \mathbb{P}^n)\geq e_P(X)e_P(Y)e_P(Z)$? Where $e_P(X)$ is the multiplicity of $P$ in $X$.

If $P$ is an isolated point, this result is true from Fulton's intersection theory book (Thm 12.4 and Cor 12.4), and it is also true for two varieties intersecting case. I want to know whether the general case is right and a reference. Thank you.

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  • $\begingroup$ In $e_P(X) e_P(Y) e_P(X)$, should the third term be $e_P(Z)$? And if $K$ is infinite, what happens if you look instead at a point in $V \cap P$ where $V$ is a generic $3$-plane? $\endgroup$ Mar 25 '14 at 16:35
  • $\begingroup$ Yes, the third one is $e_P(Z)$. I have changed it. $\endgroup$
    – var
    Mar 25 '14 at 17:35
  • $\begingroup$ Second, in your example, for a point $P$ (suppose it is closed), the multiplicity of $P$ in the closed scheme $\{P\}$ is $1$ ($\{P\}$ is equipped with the canonical reduced closed subscheme structure). And since $V$ is a plane, it is of degree $1$, then the multiplicity of $P$ in $V$ is also $1$. For the intersection multiplicity, it is also $1$. Are there any problems? $\endgroup$
    – var
    Mar 25 '14 at 17:39
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It's a result in "méthodes d'algèbre abstraite en géométrie algébrique" written by Samuel.

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