Let $X,Y\subset \mathbb P^n$ be non-empty projective varieties. I have to show that, assuming $\operatorname{dim} X+\operatorname{dim} Y\geq n$, the intersection $X\cap Y$ is non-empty.
A result that I feel like is useful is that, if $X',Y'\subset \mathbb A^n$ are pure-dimensional affine varieties, every irreducible component of $X'\cap Y'$ has dimension at least $\operatorname{dim} X'+\operatorname{dim} Y'- n$.
What I thought is: denote by $X_c$, $Y_c$ the cones in $\mathbb A^{n+1}$ corresponding to $X$, $Y$. Notice that $X_c$ is irreducible iff $X$ is. So, in case $X$, $Y$ are pure-dimensional, also $X_c$, $Y_c$ are, and the irreducible components of $X_c\cap Y_c$ all have dimension at least $\operatorname{dim} X_c+\operatorname{dim} Y_c- n-1=\operatorname{dim} X+\operatorname{dim} Y- n+1$; hence every irreducible component of $X\cap Y$ has dimension at least $\operatorname{dim} X+\operatorname{dim} Y- n$. But if $X\cap Y$ is empty, the fact that every irreducible component has dimension at least $\operatorname{dim} X+\operatorname{dim} Y- n$ is vacuously true, as there are no irreducible components. Do I just need some adjustment, or this argument won't take me anywhere? Thanks in advance
Best Answer
You still need to show that $X_c \cap Y_c$ is non-empty. But clearly, the cones intersect in $0 \in \mathbb A^{n+1}$. So $X_c \cap Y_c$ is non-empty of dimension $\geq 1$ by your dimension argument. So $X_c$ and $Y_c$ also intersect in a point $p \in \mathbb A^{n+1} \setminus \{0\}$, and $p$ projects down to a point of $X \cap Y$.