# Closed subvariety that is unique in its small analytic neighborhood

Let $$Y$$ be some smooth projective variety over $$\mathbb C$$ with $$\dim Y \geq 2$$. For a closed sub-variety $$X \hookrightarrow Y$$, consider the following property:

There is some small open neighborhood $$U$$ of $$X$$ inside $$Y$$ (in the complex topology), such that the only closed sub-variety $$Z$$ of $$Y$$ inside $$U$$ with $$\dim Z=\dim X$$ is just $$X$$ itself.

When does such $$X$$ exist (except trivial cases e.g $$X=Y$$)? Can we classify all such $$X$$ ? For simplicity, one can assume $$X$$ is a divisor and smooth.

Motivation: In Accumulation of algebraic subvarieties: Near one subvariety there are many others (?), 2, one finds an example given by $$Y=Bl_O (\mathbb P^2)$$ and $$X$$ being the exceptional divisor. In general, the divisors with such property seem to be quite special, and the existence depends on the intersection theory on $$Y$$.

• Isn't this almost equivalent to the property that the point $[X] \in \mathcal{Hilb}_{Y}$ in the corresponding hilbert scheme isolated? – Saal Hardali Jul 2 '19 at 8:05