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Let $k$ be an algebraically closed field, and $Z,U$ be smooth varieties over $k$. A variety is by definition integral (irreducible + connected).

Can we classify all smooth projective varieties $X$ over $k$ with a closed subvariety isomorphic to $Z$ and the complement isomorphic to $U$ (assume there is one)? Of course they are all birational and are the same in the Grothendieck group. In general, it seems too difficult. I hope there is a simple condition for uniqueness, and classification in low dimension case (for curve it's trivial).

Examples:

  • $Z$ is a point, this can be thought as whether the one-point compactification is unique.
  • $Z=\mathbb P^{n-1}$, $U=\mathbb A^n$, $X=\mathbb P^n$. Are there any other examples?
  • $Z=\mathbb P^{n-1}$, $U=\mathbb P^1 \times \mathbb A^{n-1}$, then consider any rank $2$ projective bundle over $\mathbb P^{n-1}$. Are there any other examples?
  • $X_0$ is an abelian variety, $Z$ an abelian subvariety, $U=X_0-Z$.
  • The general type case.
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    $\begingroup$ In another words you are asking about classifications of "projective compactifications'' of $U$. It is not even known for your second example $Z = \mathbb{P}^{n-1}$, $U = \mathbb{A}^n$. For this example the complete answer for $n = 2$ and $k = \mathbb{C}$ can be found from this paper of mine: arxiv.org/abs/1307.5577 (combine Proposition 3.3 and Corollary 3.5). In particular there are examples other than $X = \mathbb{P}^2$, e.g. the weighted projective surfaces of the form $\mathbb{P}^2(1,p,q)$. $\endgroup$
    – pinaki
    Commented Jan 5, 2020 at 3:00

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