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I learned Bezout's Theorem in class, stated for plane curves (if irreducible, sum of intersection multiplicities equals product of degrees). What is the proper general statement, for projective varieties of degree n?

I think it is something like: If finite, the sum of multiplicities equals the product of degrees.. else the (dimension? degree? sums over irreducible components?) of the intersection is less than or equal to the difference in degrees.

Help is appreciated!

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The intersection does not have to be finite to have a nice statement. You may want to try "Algebraic Geometry, a first course" by Harris which strikes a good balance between giving a somewhat general result and keeping it accessible. (Fulton is the go to reference for the full general case, but as mentioned already, it takes a whole book, so there is no short answer in that case.) –  Thierry Zell Nov 1 '10 at 12:33
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2 Answers 2

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Dear unknown, the most straightforward generalization of Bézout's theorem might be the following. Consider $\mathbb P^n $, projective space over the field $k$, and $n$ hypersurfaces $H_1,...,H_n$in general position in the sense that their intersection is a finite set. Then, calling $h_i$ their local equations, Bézout says

$$\sum dim_k \mathcal O_{\mathbb P^n,P_i}/(h_1,...,h_n) =\prod deg (H_i) $$

The dimension on the left hand side is, of course, to be interpreted as the multiplicity with which to count the point $P_i$, seen as a fat point i.e. a zero-dimensional non-reduced scheme.

A related, more abstract point of view is the description of the Chow ring of $\mathbb P^n$ as $CH^\ast (\mathbb P^n)=\mathbb Z[x]/(x^{n+1})$ ( where $x$ is the class of a hyperplane in $\mathbb P^n$). From this point of view we have the following version of Bézout. Consider $r$ cycles $\alpha_1,...,\alpha_r$ on $\mathbb P^n$ with $\alpha_i \in CH^{d_i}(\mathbb P^n) $ and $d_1+...d_r \leq n$, . Then

$$deg \prod {\alpha_i} =\prod deg (\alpha_i)$$

the product of the $\alpha_i$'s on the left being calculated in the Chow ring and the degree $deg (\alpha) $ of a cycle $\alpha \in CH^d (\mathbb P^n)$ being the integer $t$ such that $\alpha =t . x \in CH^d(\mathbb P^n)=\mathbb Z .x $.

This is only the tip of the iceberg: a definitive answer would require a book. Fortunately that book exists and has been written, to our eternal gratitude, by Fulton: Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer-Verlag, Berlin, second edition, 1998.

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In that wonderful book of Fulton, the more general result is at the bottom of page 223, where it says that for r pure dimensional schemes in P^n, whose codimensions add to at most n, the product of their degrees is at least as great as the sum of the degrees of the irreducible components of their intersection. Thus for instance, if three quadric hypersurfaces in P^4 have a common curve of degree 8, their intersection has no further components.

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A very relevant remark and example, Roy. Your statement really describes the heart of Fulton's profound but rather technical Theorem 12.3, and is formulated in a strikingly simple way ( not explicitly spelled out in Fulton), which might have escaped a reader just browsing through the book –  Georges Elencwajg Nov 2 '10 at 7:51
    
A small remark: this also holds with no restriction on the sum of codimensions (as Fulton mentions at the bottom of the same page). Just imbed everything into a larger projective space, take cones, and intersect with the complimentary subspace. –  jacob Sep 13 '12 at 6:07
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