TITLE: When is  a surface in a threefold  contractible to a curve?
QUESTION [12 upvotes]: 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?

REPLY [11 votes]: 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:

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