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:

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

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

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

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?