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 $\mathcal{Hilb}(X\times Y)$.



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> What technology can I use to decide whether $P\_y\in \mathcal{Hilb}(X\times Y)$ is smooth or not?
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Singular points on the Hilbert scheme of a product
The tangent space to the Hilbert scheme at point $P \in Hilb$ corresponding to a subvariety $Z$ is $H^0(Z,N\_Z)$, where $N\_Z$ is the normal bundle, and the obstruction space is $H^1(Z,N\_Z)$. In your case $N\_Z = T\_yY\otimes O\_X$ is a trivial vector bundle, so the tangent space is $T\_y\otimes H^0(X,O\_X)$. If $X$ is connected, it is just $T\_yY$ and it is clear that all obstructions vanish (since there is a family of deformations with the same tangent space). So, in this case $P\_y$ is a smooth point of the Hilbert scheme.


