# How to show the blowing down a ruled surface is projective

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?

• 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. – Tabes Bridges Aug 12 '20 at 0:13
• 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). – AG learner Aug 22 '20 at 14:52
• @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. – user41650 Aug 22 '20 at 22:09
• @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. – AG learner Aug 22 '20 at 22:50
• @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 – user41650 Aug 22 '20 at 23:25