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Is there a version of Bézout's theorem for non-proper intersections?

For my specific problem, the setup is as follows: Let $P_1,P_2,P_3,P_4\in\mathbb{C}[z_1,z_2,z_3,z_4]$, and suppose that (as a scheme) $\bigcap_{j=1}^4 \mathbf{Z}(P_j)$ is a 1-dimensional variety in $\mathbb{C}^4,$ plus some embedded points. I would like to say that the number of embedded points is at most $\prod_{j=1}^4 \operatorname{deg}(P_j)$. Does anyone have a reference / counter-example for such a statement?

I would be happy to lose a constant as well; a bound of $O\Big(\prod_{j=1}^4 \operatorname{deg}(P_j)\Big)$ would be fine too.

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    $\begingroup$ You would get what you want if you can show that all the embedded points are "distinguished components" of the intersection in the sense of Fulton. See the refined Bezout theorem in Chapter 12 of his "Intersection Theory". $\endgroup$
    – naf
    Commented May 8, 2013 at 5:29
  • $\begingroup$ Thanks for this, Ulrich! Do you know if an embedded point being a "distinguished component" has a simple geometric description if my variety is an intersection of four hypersurfaces in $\mathbb{C}^4$? I took a look at Fulton, but it will take me quite some time to understand all of the language in it. $\endgroup$
    – Josh Zahl
    Commented May 8, 2013 at 7:22
  • $\begingroup$ I don't know. If I have time to think about it I will add another comment later. $\endgroup$
    – naf
    Commented May 12, 2013 at 9:06

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