Let $X, Y$ be projective varieties in $\mathbb{P}^n$ for $n>10$. Assume that dimensions of $X,Y$ are greater than $n/2$. My first question is as follows: Is there any criterion (other than the definition) which tells us when $X$ and $Y$ will intersect properly?
Suppose that $X, Y$ do not intersect properly. Is there any general way of checking whether/when the intersection of $X$ and $Y$ (by intersection we mean the fiber product $X \times_{\mathbb{P}^n} Y$ followed by the pull-back by the diagonal morphism) is a non-reduced scheme?
