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Let $Y \subset P^n(\Bbb{C}$) is an irreducible projective algebraic set, then how to show that dim$Y$ is equal to the minimum $r \in \Bbb{ N }$ such that there exists a linear subspace $S_{n-r-1} \subset P^n(\Bbb{C}$) with $S_{n-r-1}\cap X=\emptyset$. It will be helpful if i get a reference for this fact.

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  • $\begingroup$ To answer this question, first you must specify which definition of dimension you are using. $\endgroup$
    – user64687
    May 7, 2014 at 15:35
  • $\begingroup$ my definition of dimension of X is the largest length of chain of irreducible closed subsets of X. $\endgroup$ May 7, 2014 at 15:41
  • $\begingroup$ Dear abc, I think this should be discussed in section 7 of Ch. I of Hartshorne (which discusses some basic results on intersections, degrees, and so on). Regards, $\endgroup$
    – Matt E
    May 9, 2014 at 1:55

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This is Lemma 3.2 of chapter VII (page 122) of Algebraic Geometry, An Introduction by Perrin.

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