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The following is a modern, fairly general form of Bézout's theorem. (There are forms that are more general and/or more precise; bear with me.)

Define the degree of a reducible variety to be the sum of the degrees of its irreducible components. Then, for any two varieties $V_1$, $V_2$ in affine space $A^n$ (or projective space $P^n$), $$\deg(V_1\cap V_2) \leq \deg(V_1) \deg(V_2).$$

I've seen this result credited to Fulton-Macpherson (e.g., in Shokurov-Danilov). This feels too late; I imagine they are really responsible for a more general result. At the same time, I haven't seen this result clearly stated earlier in the literature, not that that means anything - I am no algebraic geometer. (As a friend just reminded me, Thm. I.7.7 in Hartshorne gives the special case of $V_2$ a hypersurface; I do not see how the general case follows from that.)

So, whom should we credit the result to? What would be a proper reference?

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    $\begingroup$ A friend me tells me that Hilbert and Serre deserve credit. I'd love to give it - and I would also like to get to the bottom of the issue! $\endgroup$ Sep 20, 2022 at 8:21
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    $\begingroup$ This is example 8.4.6 in Fulton's Intersection Theory. It might be worthwhile to take a look at the Notes and References section of chapter 8, there are named many people who worked on versions of Bezout's theorem. $\endgroup$ Sep 20, 2022 at 15:35
  • $\begingroup$ In the light of Fulton's remarks (see D. Dona's reply below), would the following be best? "This is a generalization of Bézout's theorem for curves, due to Fulton and Macpherson (1980) ([Fulton, Ex. 8.4.6]; see also [Vogel]. The special case where $V_2$ is a hypersurface is classical [Hartshorne, Thm. I.7.7]." $\endgroup$ Sep 22, 2022 at 8:48
  • $\begingroup$ The funny thing is that, in the introduction of \S I.7 of Hartshorne, one can find (a) the general claim (as an inequality, with multiplicities) (b) the statement that "The hardest part of this generalization is the correct definition of intersection multiplicity", and that that was done by Severi (geometrically) and Chevalley and Weil, (c) a promise that, while \S I.7 treats only the case of $V_2$ a hypersurface, the general case will be treated in Appendix A... but then Appendix A seems to drop the ball, or at least I can't find generalized Bézout there. $\endgroup$ Sep 22, 2022 at 8:51
  • $\begingroup$ It's not clear to me if Hartshorne intended to claim a general case of Bézout's theorem was proven (which would mean that it predates Fulton's work in the early 80s), or if he was simply expecting such a theorem. He writes carefully "we should have...". About the appendix: I think Hartshorne meant a general definition of intersection multiplicity is given in the appendix (p.427), not a generalized Bézout. $\endgroup$ Sep 22, 2022 at 12:57

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I'll expand on the comment by @red_trumpet. Quoting from p. 152 of Fulton's book: "The result of Example 8.4.6 was discovered and proved with MacPherson and Lazarsfeld, in answer to a question of Kleiman".

Vogel's Lectures on Results on Bézout's theorem has some more information. Corollary 2.26 ("refined Bézout's theorem") asserts that, given pure-dimensional projective varieties $V_{i}$, the sum of the degrees of the irreducible components of $\bigcap V_{i}$ is bounded by the product of the $\deg(V_{i})$: this is the projective case of our question, essentially. The result is again credited to Fulton and MacPherson, in answer to Kleiman's 1979 question given as Corollary 2.27 (the same as above, but with "the number of the irreducible components" instead of "the sum of the degrees of the irreducible components").

Again according to Vogel, the first proof is contained in Fulton's Intersection Theory (not the 1984 book, but the 1980 notes for a summer school in Cortona), then a second proof was suggested based on Deligne's ideas, and then a new reinterpretation came from Lazarsfeld (1981). I haven't personally looked at any of these sources.

Given that Kleiman had asked a weaker question in 1979, the result seems to be in fact as recent as the book, plus/minus a few years.

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