Dimension of irreducible variety

Question

Why is the dimension of intersection, $V\cap H$, of $m$-dimensional irreducible variety $V$ and a hyperplane given by $\dim(V\cap H)$ of dimension $m-1$?

Answer 1

It doesn't always have to be $m-1$, but it must be at least dimension $m-1$. This is essentially the content of Krull's principal ideal theorem together with the fact that you for a finite type $k$-algebra $A$, one has that $\dim A/\mathfrak{p}+\text{ht}(\mathfrak{p})=\dim A$.