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Let $X \subseteq \mathbb{P}^n$ be a projective variety. I would like to have a morphism $f: \tilde{X}\to X \subseteq \mathbb{P}^n$ where $f$ is finite and birational, $f^* \mathcal{O}_{\mathbb{P}^n}(1)$ is a very ample line bundle and $\tilde{X}$ is a smooth projective variety. Under which conditions does such a map exist? I think it should always exist if $X$ is an irreducible curve, but are there other classes? What happens if we drop one of the requirements on $f$?

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up vote 5 down vote accepted

In other words, you want the normalization of $X$ to be smooth, plus a condition on the embedding: the pull back of $\mathcal{O}_X(1)$ to $\tilde{X} $ should be very ample. The latter property depends on the embedding and is not always true, even for curves: see this post and the answers there.

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Thank you for this quick answer! –  Hans Jun 26 at 15:05

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