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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?

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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.

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First, rigidity lemma: let $X\times Y \leftarrow Z\to T$ be a flat family of closed subschemes of $X\times Y$, with the fiber $Z_0$ over $0\in T$ equal to $X\times {y}$. Then the composition $Z\to X\times Y \to Y$ contracts $Z_0$ to a point, therefore it has to contract nearby fibers by the Rigidity Lemma.

I omitted a considerable amount of details here, but it should follow that the connected component of the Hilbert scheme corresponding to $P_y$ is isomorphic to $Y$, hence smooth.

A good reference for the Rigidity Lemma(s) is Mumford's book on abelian varieties, chapters 2 and 3.

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