Let $X$ and $Y$ be smooth projective varieties, say over $\mathbb C$. Fixing a point $y\in Y$, we obtain a smooth, closed subvariety $X\times\{y\}$ of $X\times Y$, which in turn corresponds to a point $P_y$ on the Hilbert scheme $\mathcal{Hilb}(X\times Y)$.

What technology can I use to decide whether $P_y\in \mathcal{Hilb}(X\times Y)$ is smooth or not?