# 1.11(b)

In Groupes de Brauer II, Remark 1.11(b) Grothendieck notes that results of Mumford’s paper “The topology of normal singularities of an algebraic surface and a criterion for simplicity” gives one an example of a normal surface Y over the complex numbers such that H^2(Y, G_m) isn’t torsion and does not inject into H^2(C(Y), G_m). Grothendieck even references a page number, namely 16. To explain this in the graduate student seminar on Brauer groups this semester I came up with the following, which may be what Grothendieck had in mind.

Let E ⊂ P^2 be a smooth degree 3 curve. Let P ∈ E be a flex point. Blow up P exactly 10 times on E, i.e., blow up P in P^2, then blow up P on the strict transform of E, etc. The result is a surface X with an embedding E ⊂ X such that

1. the self square of E in X is -P, and
2. the image of the map Pic(X) —> Pic(E) is contained in ZP.

This means you can blow down E on X to get a normal projective surface Y with a unique singular point y. Part 2 implies that the local ring of O_{Y, y} is factorial (this is one of Grothendieck’s claims — in fact we won’t need it). Now look at the Leray Spectral Sequence for G_m and the morphism f : X —> Y. You get something like

Pic(X) —> H^0(Y, R^1f_*G_m) —> H^2(Y, G_m) —> H^2(X, G_m)

We have R^1f_*G_m = Pic(E) placed at y and H^2(X, G_m) = 0 as X is a smooth projective rational surface. Using 1 and 2 above we conclude that H^2(Y, G_m) = E as abelian groups. By Gabber’s result on Brauer groups of quasi-projective schemes it follows that Br(Y) = E_{tors}. Of course both H^2(Y, G_m) and Br(Y) map to zero in the Brauer group of the generic point.