{"id":4010,"date":"2015-04-22T19:41:02","date_gmt":"2015-04-22T19:41:02","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=4010"},"modified":"2015-04-22T19:41:02","modified_gmt":"2015-04-22T19:41:02","slug":"2-and-3","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4010","title":{"rendered":"2 and 3"},"content":{"rendered":"<p>Here is a question I asked myself yesterday: Suppose that X is an algebraic space which has degree 2 finite etale covering X_2 &#8212;> X and a degree 3 finite etale covering X_3 &#8212;> X such that both X_2 and X_3 are schemes. Is X a scheme?<\/p>\n<p>I thought there might be a chance that the answer is yes, but just now in the common room, Philip, Anand, Davesh, and me proved that the answer is: no!<\/p>\n<p>Namely, following Hironaka&#8217;s example, we made a smooth proper 3-fold X_6 with an action of a cyclic group G of order 6 such that X = X_6\/G is not a scheme, but the two intermediate quotients X_2 and X_3 are schemes. Namely, start with a smooth projective 3-fold Y with an action of G and a 6-gon of smooth rational curves C_0, C_1, C_2, C_3, C_4, C_5 which are cyclically permuted by G. In other words, I mean that C_i &cap; C_{i + 1} is exactly one point P_i and the intersection is transversal. Then you do as in the Hironaka example: blow up all these curves but at P_i analytic locally blow up C_i first and C_{i + 1} second. The result is a proper scheme X_6 with an action of G. Over the P_i there are two curves E_i, E&#8217;_i such that E_0 + E_1 + &#8230; + E_5 is zero in the Chow group (see picture of Philip below). Hence orbit of a point on E_0 cannot be contained in an affine open of X_6, which proves that X_6\/G is not a scheme. However, the morphism<\/p>\n<blockquote><p>X_6 &#8211; E_i &cup; E&#8217;_i &#8212;-> Y &#8211; P_i<\/p><\/blockquote>\n<p>is a usual blow up, hence the source is a quasi-projective variety and any finite collection of points is contained in an affine open. This quickly gives that X_3 and X_2 are schemes.<br \/>\nTo finish, here is a picture X_6 (thanks Philip!):<a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/wp-content\/uploads\/2015\/04\/IMG_20150422_153508_334.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/wp-content\/uploads\/2015\/04\/IMG_20150422_153508_334.jpg\" alt=\"IMG_20150422_153508_334\" width=\"2448\" height=\"3264\" class=\"aligncenter size-full wp-image-4011\" srcset=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/wp-content\/uploads\/2015\/04\/IMG_20150422_153508_334.jpg 2448w, https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/wp-content\/uploads\/2015\/04\/IMG_20150422_153508_334-225x300.jpg 225w, https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/wp-content\/uploads\/2015\/04\/IMG_20150422_153508_334-768x1024.jpg 768w\" sizes=\"auto, (max-width: 2448px) 100vw, 2448px\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here is a question I asked myself yesterday: Suppose that X is an algebraic space which has degree 2 finite etale covering X_2 &#8212;> X and a degree 3 finite etale covering X_3 &#8212;> X such that both X_2 and &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4010\">Continue reading <span class=\"meta-nav\">&rarr;<\/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-4010","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\/4010","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=4010"}],"version-history":[{"count":3,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4010\/revisions"}],"predecessor-version":[{"id":4014,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4010\/revisions\/4014"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4010"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4010"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4010"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}