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An automorphism is an isomorphism from an object to itself, and which also preserves the objects structure .

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When does $\operatorname{Aut}(X)=\operatorname{Bir}(X)$ hold?

To complete the answer of Diverietti and the comment of Roy Smith, here is a statement which might interest you: Theorem If $X$, $Y$ are varieties over a field $k$, assume $X$ is smooth and $Y$ proper …
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