Let $X$ be an irreducible projective threefold, let $S$ be a ruled surface over a curve $C$, where $C$ has finitely many singular points. Let $\pi:X\rightarrow Y$ be a map such that $\pi(S)=C\subset Y$ and $\pi$ is isomorphism outside $S$. The fiber $F\cong\mathbb{P}^1$ of $\pi$ over each point $P$ on $C$ is smooth if $P$ is smooth and it is singular if $P$ is singular. $X$ is smooth outside $S$, and $\pi$ is etale outside $S$. Also I know $Y$ is irreducible three dimensional and is smooth outside $C$.

Is there a way to show that $Y$ is also projective?

  • $\begingroup$ What you're looking for is something to do with Fujiki's contractibility criterion. I don't have a precise statement on hand, but it has to do with the normal bundle to $S$ being suitably negative ala Castelnuovo's criterion. $\endgroup$ – Tabes Bridges Aug 12 '20 at 0:13
  • $\begingroup$ Even if $C$ is smooth, there are examples that the contraction $Y$ is not projective. It will be good to know if there is a projectivity criterion (though I didn’t find any). $\endgroup$ – AG learner Aug 22 '20 at 14:52
  • $\begingroup$ @AGlearner I think if $C$ is smooth, then we are talking about Mori contraction, then by Cone theorem, one can show the contraction is projective. $\endgroup$ – user41650 Aug 22 '20 at 22:09
  • $\begingroup$ @user41650 Let's say $X$ is a smooth threefold with a rational curve $C$ with normal bundle $N_{C|X}=\mathcal{O}(-1)+\mathcal{O}(-1)$, then blow up $X$ along $C$ is a surface $S$ identified with $\mathbb P^1\times \mathbb P^1$ with normal bundle $\mathcal{O}(-1,-1)$ (i.e., restriction to both rulings have $\mathcal{O}(-1)$ direction), so one can contract $S$ along the other ruling, and get a smooth threefold $X'$. This is flop, a flop of projective variety does not need to be projective. $\endgroup$ – AG learner Aug 22 '20 at 22:50
  • $\begingroup$ @AGlearner Sorry I am not an expert in birational geometry, but is what you said contradicts to Mori contraction for smooth projective threefold? I am contracting a ruled surface to a smooth curve and every curve I contract is a $K_X$-negative extremal ray, then there are only one possibility, the contraction of $X$ is a projective variety $\endgroup$ – user41650 Aug 22 '20 at 23:25

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