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Let $X$ be a Noetherian irreducible scheme of dimension $n$. Let $Y,Z$ be its closed irreducible subschemes of dimensions $k,l$ respectively.

Under what technical conditions the dimension of each irreducible component of $Y\cap Z$ is at least $k+l-n$?

Say is that true for any regular (variety over a field) $X$ ?

Remark. In Propositions I.7.1 and I.7.2 in Hartshorne's book this property is proven in the cases when $X$ is either affine space $\mathbb{A}^n$ or projective space $\mathbb{P}^n$ over a field.

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This is true for $X$ a non-singular variety, see Fulton, Intersection Theory, section 8.2, p. 137.

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    $\begingroup$ however on the singular quadric 3-fold xy=zw, the planes x=z=0 and y=w=0 meet only at the point (0,0,0,0). Mumford's redbook chapter 3.6. $\endgroup$
    – roy smith
    Jun 14, 2017 at 4:07

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