{"id":4115,"date":"2016-03-01T18:05:53","date_gmt":"2016-03-01T18:05:53","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=4115"},"modified":"2016-03-07T11:48:55","modified_gmt":"2016-03-07T11:48:55","slug":"blowing-down-exceptional-curves","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4115","title":{"rendered":"Blowing down exceptional curves"},"content":{"rendered":"

Let X be a Noetherian separated scheme. Let E ⊂ X be an effective Cartier divisor such that there is an isomorphism E → P<\/b>1<\/sup>k<\/sub> where k is a field. Then we say E is an exceptional curve of the first kind if the normal sheaf of E in X has degree -1 on E over k.<\/p>\n

You can get an example of the situation above by starting with a Noetherian separated scheme Y and a closed point y such that the local ring of Y at y is a regular local ring of dimension 2 and taking the blowup b : X → Y of y and taking E to be the exceptional divisor.<\/p>\n

Conversely, if E ⊂ X is gotten in this manner we say that E can be contracted.<\/p>\n

The following questions have been bugging me for a while now.<\/p>\n

Question 1:<\/b> Given an exceptional curve E of the first kind on a separated Noetherian scheme X is there a contraction of E?<\/p>\n

Question 2:<\/b> Given an exceptional curve E of the first kind on a separated Noetherian scheme X is there a contraction of E but where we allow Y to be an algebraic space?<\/p>\n

Question 3:<\/b> Suppose that Y is a separated Noetherian algebraic space and that y is a closed point of Y such that the henselian local ring of Y at y is regular of dimension 2. Is there an open neighbourhood of y which is a scheme?<\/p>\n

Question 4:<\/b> With assumptions as in Question 3 assume moreover that the blow up of Y in y is a scheme. Then is Y a scheme?<\/p>\n

In these questions the answer is positive if we assume that X or Y is of finite type over an excellent affine Noetherian scheme (and I think in the literature somewhere; I’d be thankful for references).<\/p>\n

But… it might be interesting and fun to try and find counter examples for the general statements. Let me know if you have one!<\/p>\n","protected":false},"excerpt":{"rendered":"

Let X be a Noetherian separated scheme. Let E ⊂ X be an effective Cartier divisor such that there is an isomorphism E → P1k where k is a field. Then we say E is an exceptional curve of the … 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-4115","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\/4115","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=4115"}],"version-history":[{"count":6,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4115\/revisions"}],"predecessor-version":[{"id":4121,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4115\/revisions\/4121"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4115"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4115"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4115"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}