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I'm confused about corollary 11.13 (see e.g. google books) in the afforementioned book, namely its second half (the dimension formula). If we take $X$ to be the disjoint union of a point and a line, $\pi$ any constant map on $X$, wouldn't $X_0=point$ give a counterexample?

The statement reads: Let $X$ be a projective variety and $\pi:X \rightarrow P^n$ any regular map; let $Y$ be its image. For any $q\in Y$, define $\lambda(q)= dim (\pi^{-1}(q))$. Then $\lambda$ is an upper semicontinuous function of $q$.

Moreover, if $X_0 \subset X$ is any irreducible component, $Y_0$ its image under $\pi$, and $\lambda$ the minimal value of $\lambda(q)$ on $Y_0$, we have $$dim(X_0)=dim(Y_0)+\lambda$$

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could you reproduce the statement your think is refuted by your example? google books doesn't show the second half of the corollary to me. –  Dima Sustretov Aug 7 '12 at 12:51
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yes, $\lambda$ should be taken to be the minimum of $\dim(\pi^{-1}(q) \cap X_0)$ on $Y_0$ for the equality to hold. –  Dima Sustretov Aug 7 '12 at 14:41
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1 Answer 1

Yes, Harris is wrong and your counter-example is right.
Moreover I just checked my copy of the book and saw that I had scribbled a criticism of that corollary and its proof (I don't remember when).
A correct treatment of these questions can be found in Perrin's Algebraic Geometry, Chapter IV, section 3 d, pages 80-82.

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there is also a good discussion of dimension formula in Ravi Vakil's course notes, available here: math.stanford.edu/~vakil/216blog Chapter 12, Section 4 –  Dima Sustretov Aug 7 '12 at 14:34
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