In shafarevich's book "Basic algebraic geometry", there is an intersection dimension theorem as follows:

Let $V$, $W$ be any two irreducible closed subvarieties of $\mathbb{A}^n$ (the affine space over an algebraic closed field $k$), if $V\cap W\neq \emptyset$ , then each irreducible component of $V\cap W$ has dimension at least $\dim(V)+\dim(W)-n$.

My question is if we replace $\mathbb{A}^n$ by any smooth variety, is the statement still true? Is there any reference?

PS: I know a counterexample if $\mathbb{A}^n$ is replaced by singular variety.

  • 2
    $\begingroup$ This is true for any smooth variety. You can imitate the proof in the case of subvarieties in an affine space. The crucial facts are when $X$ is a smooth variety, then the diagonal $\Delta(X)$ in $X\times X$ is locally a complete intersection, and also that the dimension of irreducible algebraic varieties can be computed locally. The result should be somewhere is Fulton's "Intersection theory", but I was not able to find within one minute. $\endgroup$ – Cantlog Mar 26 '14 at 17:43

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