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Given a threefold $Y$ containing a surface $S$. Under which conditions can I contract $S$ so that I still end up with a smooth variety? In other words

what are the conditions for the existence of a smooth variety $X$ and a morphism $Y\rightarrow X$ such that the image of $S$ under the morphism is a curve and the morphism is an isomorphism away from $S$?

What are the conditions when $Y$ is a fourfold and $S$ is still a surface?

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up vote 10 down vote accepted

You want a divisorial contraction $Y \to X$ on a smooth 3-fold $Y$. Extremal divisorial contractions (i.e., contractions associated to a $K_Y$-negative extremal ray in the Mori cone of $Y$) have been classified by S. Mori in his paper

3-folds whose canonical bundles are not numerically effective, Annals of Mathematics 116, No. 1 (1982), pp. 133-176.

Looking at Theorem 3.3, we see that here are exactly the following possibilities:

  1. the smooth blow-up of a point; in this case $S$ is isomorphic to $\mathbb{P}^2$ with normal bundle $\mathcal{O}(-1)$;

  2. the smooth blow-up of a curve, in this case $S$ is a ruled surface whose normal bundle restricted to the ruling has degree $(-1)$; this is the situation described by Donu Arapura in his answer;

  3. the contraction of a plane $S$ with normal bundle $\mathcal{O}(-2)$; in this case the surface $X$ has an isolated singularity isomorphic to the quotient of $\mathbb{A}^3$ by the involution $(x,y,z) \to (-x, -y, -z)$;

  4. the contraction of a smooth quadric $S$ whose rulings are numerically equivalent; in this case the image of $S$ is a single point, which is a singular point for $X$;

  5. the contraction of a singular quadric $S$; again, the image of $S$ is a point in $X$.

Summing up, if you want that $X$ is smooth and the image of $S$ is a curve, the only possibility is 2.

In the case where $Y$ is a smooth 4-fold and $S$ is a smooth surface, the answer can be found in the paper of Kawamata "Small contractions of four-dimensional algebraic manifolds": in this case the only possibility is that $S$ is the disjoint union of copies of $\mathbb{P}^2$, with normal bundle $\mathcal{O}(-1) \oplus \mathcal{O}(-1)$

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@ Francesco. Thank you very much for your answer. In the of a smooth 4-fold, Kawamata's paper assumed that the contraction is " a small elementary contraction". It sounds that this is a pretty strong restriction since there are no small elementary contractions in the case of algebraic 3-folds. Do you know if there are any results with weaker assumptions? – JME Aug 14 '10 at 18:52
Unfortunately I'm not aware of any results with weaker assumptions. Maybe they exist, but I'm not really a specialist in Mory theory... – Francesco Polizzi Aug 18 '10 at 9:43
@FrancescoPolizzi: I just saw this old answer. There is something I do not quite understand; maybe I am missing something simple. Mori's theorem classifies $K$-negative extremal contractions, so all the curves contracted are numerically linearly dependent. How do we know that contractions satisfying the OP's conditions will be extremal? (The kind of situation I have in mind is that we contract a conic bundle with a reducible fibre, and the two components of that fibre are numerically independent.) – user5117 Nov 7 '14 at 15:48
@ArtiePrendergast-Smith: Right, my answer only considers extremal contractions. I corrected it. Thanks for the remark. – Francesco Polizzi Nov 17 '14 at 9:07

You want a divisorial contraction. This paper may be the answer.

  1. MR2041612 (2005c:14019) Tziolas, Nikolaos . Terminal 3-fold divisorial contractions of a surface to a curve. I. Compositio Math. 139 (2003), no. 3, 239--261.

from the paper "This paper studies divisorial contractions of a surface to a curve, i.e. when dim $\Gamma = 1$ and X has only index 1 terminal singularities along $\Gamma$. It is not always true that given $\Gamma \subset X$, there is a terminal contraction of a surface to $\Gamma$. We investigate when there is one, give criteria for existence or not and in the case that there is a terminal contraction we also describe the singularities of Y."

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Do you have a reference for the case where $X$ and $Y$ are smooth fourfolds and $S$ is still a surface contracting to a curve? – JME Jul 11 '10 at 22:55
@JME, I can't access these two papers right now but see (MR1620110, MR1638131). "...They achieve in dimension four what Mori completed for threefolds. In this they constitute the first significant step of the last decade in the four-dimensional minimal model program..." If not, ask Donu Arapura ! – SandeepJ Jul 12 '10 at 13:11

I think there are birational geometers lurking around who would do a better job. But let me make a small attempt for now. If you blow up a smooth curve on a smooth threefold, you would get a ruled surface with normal bundle restricting to $O(-1)$ along the rulings. I think you can do the converse if you allow yourself to work in the category of smooth algebraic spaces [see Artin, Cor. 6.11, Algebraization of formal moduli..., Annals 1970], but such a statement is generally be false for varieties. I remember learning this from Moishezon long ago.

I think that if in addition to the above conditions, the fibre of the ruling is an extremal ray in Mori's sense, you can probably use his stuff to get a projective contraction. Take a look at Kenji Matsuki's book on the Intro. to the Mori program for more about that.

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I hadn't realized there was already an answer, but I'll let mine stand for now. – Donu Arapura Jul 11 '10 at 22:07
Your reference is very useful! Thank you! – JME Jul 11 '10 at 23:19

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