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To improve my chances of getting answers/ comments I post my mathstackexchange question https://math.stackexchange.com/q/2808852/42548 also here.

I am trying to learn a bit about birational morphisms: $f:X→Y$, between (projective) normal varieties.

In particular, it is well known that every such morphism is a blow-up (e.g Hartshorne, Algebraic Geometry Theorem 7.17)

Suppose $f:X\rightarrow Y$ is a contraction, i.e., f has connected fibers. The situation when the exceptional set of $f$ has codimension 1 (divisorial contraction) is very different from the case when the exceptional set has codimension greather than or equal to two (small contraction).

In particular, $f$ can only be a small contraction if Y is not $\mathbb{Q}$-factorial (e.g., Kollar, Mori, Birational Geometry , Corollary 2.63). I am wondering about the converse, i.e., suppose that Y is not $\mathbb{Q}$-factorial is it then true that there exists an X as above and a small contraction $f:X→Y$?

My naive idea is that blowing up a weil-divisor which is not $\mathbb{Q}$- Cartier "should" produce a small contraction. Is this true?

Questions:

  1. Is the blowing up at a Weil non $\mathbb{Q}$-Cartier divisor a small contraction?

  2. Is there (another) general recipe starting from a singular enough $Y$ and blowing up $Z\subset Y$ that will always give a small contraction?

  3. In general, in terms of $f$ as a blow up at Z in Y how can I tell if $f$ is small or not?

Edit: As Jason Starr pointed out in the comment below, not all surfaces are $\mathbb{Q}$-factorial. Since there are no small contractions to a surface my first question is trivially false in general. However, I would love to hear some comments on when a higher dimensional normal non $\mathbb{Q}$-factorial variety is the target of a small contraction. Maybe, someone could say something about question 2 and 3 in higher dimension for some special type of varieties?

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    $\begingroup$ Every projective cone $Y$ in $\mathbb{P}^3$ over a smooth curve $C$ of degree $d\geq 3$ in $\mathbb{P}^2$ is a normal, projective variety that is not $\mathbb{Q}$-factorial (the Picard group of $Y$ is just $\mathbb{Z}$, but the Weil divisor class group contains the Jacobian of $C$). Since $Y$ is a surface, every birational proper morphism $f:X\to Y$ has codimension $1$ exceptional set. So $f$ is not a small contraction. $\endgroup$
    – Jason Starr
    Commented Jun 6, 2018 at 11:46
  • $\begingroup$ Thank you @Jason Starr for explaining that there are surfaces which are not $\mathbb{Q}$-factorial. If you write your comment as an answer I would accept it since it answers my question. However, I also added an edit to my question in order to get some more insights on when a higher dimensional non $\mathbb{Q}$-factorial variety is the target of a small contraction and how one maybe could go about producing small contractions by blowing-up. I would love to hear about what is known in this direction. If you like to share any insights I would be grateful. Thank you very much for your time. $\endgroup$
    – harajm
    Commented Jun 6, 2018 at 15:26
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    $\begingroup$ @harajm: Take a product of a non $\mathbb{Q}$-factorial surface with something smooth. This will give you an example in dimension higher than 2. $\endgroup$
    – Sasha
    Commented Jun 12, 2018 at 18:13
  • $\begingroup$ As I mentioned in an answer to mathoverflow.net/a/45193/10076, there is a pretty good way to understand what happens if $Y$ has klt singularities (incidentally, comparing to Jason's comment, a cone over a curve in $\mathbb P^2$ is klt iff its degree is at most $2$). What happens is that in that case there is a small contraction from a $\mathbb Q$-factorial model, so the existence of a small resolution depends on whether that (non-unique) model can be smooth. $\endgroup$
    – Sándor Kovács
    Commented Mar 29, 2019 at 7:39

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