Small contractions as blow ups I am trying to learn a bit about birational morphisms: $f: X\rightarrow Y$, 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 $f$ is a contraction, i.e., $f$ has connected fibers. The situation when the exceptional set of $f$ has codim $1$ 
(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 $f:X\rightarrow Y$?
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? 
 A: In general, the blow-up along a Weil divisor that is not $\mathbb Q$-Cartier does not need to be a small contraction. Indeed, assume that $S$ is a surface that is normal but not $\mathbb Q$-factorial. Then, if you blow-up a prime Weil divisor $P$ that is not $\mathbb Q$-Cartier, you will obtain a birational modification along the finitely many points where $P$ is not principal. Say we get $\pi: S' \rightarrow S$. Since $S$ is normal, the fibers need to be connected. Since $P$ is turned into a Cartier divisor on $S'$, $\pi$ can not be an isomorphism. Thus, there must be some exceptional curves mapping to the bad points.
Notice that we are playing with the fact that there are no small birational morphisms between normal surfaces. Also, the above sketch works even if $P$ is $\mathbb Q$-Cartier but not Cartier. A good exercise is to do the blow-up of a line through the vertex of the cone over a plane conic.
In general, it is very useful to know that a variety $X$ is $\mathbb Q$-factorial, i.e. every Weil divisor has a multiple that is Cartier. For instance, if you have a birational morphism, it guarantees that there is an effective exceptional divisor that is relatively anti-ample (Lemma 2.62 in KM). In birational geometry there is a standard modification that is called $\mathbb Q$-factorialization: This is a small birational morphism that turns a variety with nice singularities into a $\mathbb Q$-factorial one. Let me be more precise with the following:
Assume that $(X,\Delta)$ is a klt pair, with $\Delta \geq 0$. By definition $K_X+\Delta$ is $\mathbb Q$-Cartier, but $X$ may not be $\mathbb Q$-factorial. Thus, take a log resolution $f:Y \rightarrow X$ of $(X,\Delta)$. We can write
$$
f^*(K_X+\Delta)+\sum_{j=1}^lb_jF_j=K_Y+f^{-1}_*\Delta+\sum_{i=1}^na_iE_i,
$$
where $f^{-1}_*\Delta$ denotes the strict transform of $\Delta$. The $E_i$'s and the $F_j$'s are exceptional divisors, and their union gives all of the exceptional divisors. We group them so that $a_i \geq 0$ and $b_j>0$ for all $i$ and $j$. Notice that $a_i=0$ is allowed.
What we have done so far is legal, it just depends on ordering the exceptional divisors. Also, notice that, by the klt assumption, $a_i<1$ for all $i$.
Now, fix a very small rational number $0 < \epsilon \ll 1$. Then, we can write
$$
f^*(K_X+\Delta)+\sum_{j=1}^lb_jF_j+\epsilon \sum_{i=1}^n E_i=K_Y+f^{-1}_*\Delta+\sum_{i=1}^n (a_i+\epsilon)E_i.
$$
Call $K_Y+\Delta_Y$ the right hand side. Since $Y$ is a log resolution, and $(X,\Delta)$ is klt (i.e. all $a_i<1$), then $(Y,\Delta_Y)$ is still klt.
Run an MMP for $K_Y+\Delta_Y$ relative to $X$ (see Example 2.16 in KM). Essentially, we contract the $(K_Y+\Delta_Y)$-negative curves that are exceptional over $X$. By work of Birkar, Cascini, Hacon and McKernan (http://www.ams.org/journals/jams/2010-23-02/S0894-0347-09-00649-3/S0894-0347-09-00649-3.pdf) this MMP terminates with a birational model $X'$ where $K_{X'}+\Delta_{X'}$, defined as the pushforward of $K_Y+\Delta_Y$ along the MMP, is semi-ample over $X$ (i.e. a multiple is basepoint-free relatively to $X$).
Now, relatively over $X$, we have
$$
K_Y+\Delta_Y \equiv \sum_{j=1}^lb_jF_j+\epsilon \sum_{i=1}^n E_i,
$$
where $\equiv$ denotes numerical equivalence. Thus, the MMP we ran is also an MMP for $\sum_{j=1}^lb_jF_j+\epsilon \sum_{i=1}^n E_i$. But $\sum_{j=1}^lb_jF_j+\epsilon \sum_{i=1}^n E_i$ is effective and exceptional over $X$, and therefore it is not semi-ample over $X$ (an effective and exceptional divisor is far from being semi-ample).
Notice that $\sum_{j=1}^lb_jF_j+\epsilon \sum_{i=1}^n E_i$ is supported on all of the exceptional divisors for $Y \rightarrow X$. Thus, its pushforward to $X'$ is supported on all of the divisors that are exceptional for $X' \rightarrow X$.
So, on $X'$ we have something that is semi-ample relatively to $X$, and is equivalent over $X$ to an effective divisor supported on all of the exceptional divisors for $X' \rightarrow X$.  What we conclude is that we have contracted all of the exceptional divisors for $Y \rightarrow X$, and that $X' \rightarrow X$ is small.
Since $Y$ is smooth, it is $\mathbb Q$-factorial. Also, the MMP preserves the $\mathbb Q$-factorial property (see somewhere in Chapter 3 of KM). Thus, $X'$ is $\mathbb Q$-factorial. Notice that $X'$ is called a $\mathbb Q$-factorialization and not the $\mathbb Q$-factorialization since to run the MMP we make choices, and the outcome may vary.
Notice that this tells us that all klt surfaces are $\mathbb Q$-factorial, since there are no small morphisms for surfaces.
