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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 requirementrequirements on $f$ being birational?

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 the requirement on $f$ being birational?

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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Hans
Hans
  • 3k
  • 13
  • 17

Some Kind of Resolution of Singularites

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 the requirement on $f$ being birational?