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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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3 Answers 3

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.) –  Artie Prendergast-Smith Nov 7 at 15:48
@ArtiePrendergast-Smith: Right, my answer only considers extremal contractions. I corrected it. Thanks for the remark. –  Francesco Polizzi Nov 17 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 alpha.science.unitn.it/~andreatt/bravo/rev.html (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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