Let $X_1,...,X_r\subseteq \mathbb{P}^n$ be projective varieties of dimensions $i_1,...,i_r$. Are there any criteria to determine if $X_1\cap ... \cap X_r\neq \varnothing$?
I know that if I have $X_1,X_2\subseteq \mathbb{P}^n$ and $\dim(X_1)+\dim(X_2)\geq n$, then $X_1\cap X_2\neq \varnothing$, but I don't know if this generalizes nicely in any way to an arbitrary number of varieties.
Can we tell something about hypersuperfaces?