The complement of a neighborhood of a divisor.

Question

Given a projective variety $X$ and a hyperplane section $H$ (intersection of some hyperplane in the ambient projective space with $X$). Is it true that complement of any neighborhood of $H$ is a finite number of closed points? My reasoning is that since $H$ is ample by Nakai–Moishezon it has non-zero intersection number with closed sub-varieties in $X$ that are not points. Since the complement has zero intersection with $H$, it implies it has to be points.

Answer 1

Yes, this is correct. More simply, if $X$ is a closed subvariety of $\mathbb{P}^n$, then so is any closed subvariety of $X$, so you just need to use the fact that a hyperplane in $\mathbb{P}^n$ has nontrivial intersection with any closed subvariety of $\mathbb{P}^n$ of positive dimension.