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 $f$?



Some Kind of Resolution of Singularites
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](https://mathoverflow.net/questions/163802/does-normalization-of-projective-varieties-preserve-very-ampleness) and the answers there.


