I am trying to find a projective variety $Z$ and closed subsets $X,Y \subseteq Z$ with $\dim(X)+\dim(Y) \geq \dim(Z)$ and $X \cap Y = \varnothing$.
However, all my attempts failed.
In fact, we cannot have $Z=\mathbb{P}^n$ for any $n$. If we had that, and knowing that $dim(\mathbb{P}^n)=n$, we had $\dim(X)+\dim(Y) \geq n$ and then $X \cap Y$ is non-empty (it's a result that I proved previously).
So, I tried things like the union of the graph of two parabolas that are disjoint, but this kind of things have always a common point in $\mathbb{P}^2$
Also, if $X,Y$ are points, such as $X=\{(1:0:0)\}$ and $Y=\{(0:1:0)\}$, we have $\dim(X)=\dim(Y)=0$. If we take $Z=X \cup Y$, does it work?
I'm a bit lost... maybe a hint will lead me to a correct example.
Thanks!
Edit: Of course it doesn't work. I found that $V(z^2-yz-y^2+x^2+2xy)$ is irreducible, and then is a projective variety of dimension $1$. Now I am trying to find some $X$ closed of dimension $1$...
Consider the direct product of two projective lines or, better, of a projective line and projective n-space for any n.