I asked this question a few days ago on math.stackexchange with no success, and it doesn't seem like there'll be any. So I thought I'll repost it here.
A recent big result proved by $\mathrm{\check{C}}$esnavi$\mathrm{\check{c}}$ius states that
For a regular, integral, noetherian scheme $X$ and an open subset $U \subset X$ whose complement is of codimension at least $2$, the restriction map $\mathrm{Br}(X) \rightarrow \mathrm{Br}(U)$ is an isomorphism.
This is called purity for Brauer groups. I wonder how much of the result can be extended to curves. Say, for example, we have an elliptic curve $$E: Y^2Z = X^3 + aXZ^2 + bZ^3$$ and we remove the point of origin $O$ to obtain the affine model $$C:y^2 = x^3+ax+b$$ whose complement $\{O\}$ is of codimension $1$. How much can be said about the restriction map $\mathrm{Br}(E) \rightarrow \mathrm{Br}(C)$? By a result of Bertuccioni in Brauer groups and cohomology,
Let $X$ be an separated noetherian scheme and $U \subset X$ be a nonempty open subscheme. Assume that $U$ contains every generic point and every singular point of $X$. Then the restriction map $\mathrm{Br}(X) \rightarrow \mathrm{Br}(U)$ is an injective homomorphism.
I would like to know if the example given has any chance of being an isomorphism.
