Let $X$ and $Y$ be smooth projective varieties and let $f:X - \to Y$ be a rational map defined on $U$. Define the graph of this rational map to be the closure of $\{(u,f(u)): u\in U\}$ in $X\times Y$. Is the graph necessarily smooth?
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No, see this example: https://math.stackexchange.com/q/3828093 – AG learner Sep 29 '20 at 01:21