{"id":2062,"date":"2011-11-06T18:51:26","date_gmt":"2011-11-06T18:51:26","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=2062"},"modified":"2011-11-06T18:51:26","modified_gmt":"2011-11-06T18:51:26","slug":"1-11b","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2062","title":{"rendered":"1.11(b)"},"content":{"rendered":"

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<\/strong>(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.<\/p>\n

Let E \u2282 P<\/strong>^2 be a smooth degree 3 curve. Let P \u2208 E be a flex point. Blow up P exactly 10 times on E, i.e., blow up P in P<\/strong>^2, then blow up P on the strict transform of E, etc. The result is a surface X with an embedding E \u2282 X such that<\/p>\n

    \n
  1. the self square of E in X is -P, and<\/li>\n
  2. the image of the map Pic(X) —> Pic(E) is contained in Z<\/strong>P.<\/li>\n<\/ol>\n

    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<\/p>\n

    Pic(X) —> H^0(Y, R^1f_*G_m) —> H^2(Y, G_m) —> H^2(X, G_m)<\/p><\/blockquote>\n

    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.<\/p>\n","protected":false},"excerpt":{"rendered":"

    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 … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-2062","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2062","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2062"}],"version-history":[{"count":12,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2062\/revisions"}],"predecessor-version":[{"id":2074,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2062\/revisions\/2074"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2062"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2062"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2062"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}