Dimension of the intersection between a projective variety and a hyperplane.

Question

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 $H\cap X$?

Answer 1

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.