I know there are similar questions, but everyone uses different approaches and it's complicated to change proofs. In my algebraic geometry course, we're dealing with algebraic variety (topological space locally isomorphic to Zariski closed set in affine spaces over a field K which are separated and finite type) and we defined the dimension of irreducible algebraic variety as the transcendence degree of the field of rational map (=regular maps defined on open dense sets).My question is about a proposition:
Let $X$, $Y$ be irreducible and closed subsets of a projective space $\Bbb P^n$ such that $\dim(X)+\dim(Y) \geq n$. Then every irreducible components of $X\cap Y$ has dimension at least $\dim(X)+\dim(Y)- n$.
For the proof we used the following fact
Let $X$ be an irreducible and closed subset of projective space $\Bbb P^n$ such that $\dim(X)>0$. Let $H$ be a hyperplane such that $X$ is not contained in $H$. Then $H\cap X$ is closed and each irreducible component has dimension $\dim(X)-1$.
My professor demonstrated the first proposition showing that $X\cap Y$ is isomorphic $\Delta\cap J(X,Y)$ where $\Delta$ is the set $\{[x,x]\}$ of $P^{2n+1}$ and $J(X,Y)$ is a particular irreducible algebraic variety (called the "join"). Well, $\Delta$ is just linear subspace of codimension $n+1$, i.e. intersection of $n+1$ hyperplanes. Then, the professor concluded saying "repeating the fact about intersection with hyperplane we get the thesis for each irreducible components: intersect $J$ with one hyperplane, then take the irreducible components e intersect each with another hyperplane until we get to $\Delta\cap J$".Here's the problem: is it true that taking the irreducible components of the first intersection and intersecting them with ... I get to the final irreducible components?
I don't know if it is clear my doubt. I know there are proofs that uses Krull's theorem, prime ideals, etc., but we're having a geometrical approach and I'd like to remain in that environment.
There's the chance that the proof is false, but maybe there's something I can learn. If you know some books about that has this geometrical approach, tell me please.