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Suppose that $X$ is a smooth $m$-dimensional projective variety embedded in some $\mathbb P^n_k$ (we work over an algebraically closed field). Now consider a hyperplane $H\subseteq\mathbb P^n_k$ of dimension $n-r$ with $r<n$. Is there a way to calculate the dimension of the schematic intersection $H\cap X$?

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    $\begingroup$ It is strictly forbidden to call $H$ a hyperplane if $r\neq 1$. $\endgroup$ – Georges Elencwajg Jan 17 '15 at 13:59
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If $X,Y\subset \mathbb P^n_k$ are irreducible subvarieties (smoothness is irrelevant), then for every irreducible component $Z$ of $ X\cap Y$ we have $$\operatorname {codim} Z\leq\operatorname {codim} X+\operatorname {codim}Y $$
Moreover $$\operatorname {codim} X+\operatorname {codim}Y \leq n \implies X\cap Y\neq \emptyset$$
The fact that in your case $Y=H$ is linear is irrelevant.

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