TITLE: Conditions for a smooth scheme of finite type with trivial class group to be quasi-affine
QUESTION [7 upvotes]: Let $X$ be a smooth scheme of finite type over an algebraically closed field of characteristic zero and with a trivial class group $Cl(X)=0$. Let $Y$ be a dense open subscheme of $X$ such that: 
1) $Y$ is a quasi-affine scheme s.t. $\Gamma(\mathcal{O}_Y,Y)$ is of finite type; and 
2) $X \setminus Y$ is irreducible of codimension at least two.   
Does these conditions imply that $X$ is itself a quasi-affine scheme? If not, can anyone provide a counter-example please?
Thank you in advance!

REPLY [2 votes]: Here is a counterexample. Unfortunately it is not separated, so I don't know how interesting it is to you.
Consider $X = \mathbf{A}^2 \cup \mathbf{A}^2$, glued along $\mathbf{A}^2 \setminus \{(0,0)\}$, and let $Y$ be one copy of $\mathbf{A}^2$. $X$ is not quasi-affine since it is not separated. $\operatorname{Cl}(X) \cong \operatorname{Cl}(Y) = 0$ since the complement of $Y$ in $X$ is a point, which has codimension 2. Also, $Y$ is affine, and $\Gamma(Y,\mathcal{O}_Y) = k[x,y]$ is of finite type.
Finally, some remarks which I commented earlier:

Schröer has an example of a complete normal variety with $\operatorname{Pic}X = 0$, but I didn't compute its class group since you wanted a smooth example.
Hamm and Lê show that a complex algebraic variety with $H^1 = H^2 = 0$ (the actual condition is weaker) would have trivial class group, so this might be a place where you could find a counterexample that is also a variety.