{"id":717,"date":"2010-08-08T15:55:01","date_gmt":"2010-08-08T15:55:01","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=717"},"modified":"2010-08-08T15:55:01","modified_gmt":"2010-08-08T15:55:01","slug":"closed-points-in-fibres","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=717","title":{"rendered":"Closed points in fibres"},"content":{"rendered":"<p>Yesterday I found this in a <a href=\"http:\/\/arxiv.org\/abs\/1007.1314v1\">preprint<\/a> by Brian Osserman and Sam Payne:<\/p>\n<ul>\n<li>If X &#8212;&gt; S is locally of finite type, and x -&gt; z is a specialization of points in X with z a closed point of its fibre, then there exist specializations x -&gt; y, y -&gt; z such that y is in the same fibre as x and is a closed point of it. Moreover, the set of all such y is dense in the closure of {x} in its fibre.<\/li>\n<\/ul>\n<p>I was already planning to try to prove this and add it to the stacks project as I think that it could be quite useful.<\/p>\n<p>To prove this statement you first reduce to the case where the base is a valuation ring and the morphism is flat. My idea was to use an argument a la Raynaud-Gruson to reduce to the case of a smooth morphism, where you can slice the map, i.e., argue by induction on the dimension. Brian and Sam&#8217;s argument is simpler: they show that you can do the slicing without reducing to a smooth morphism by showing that a locally principal closed subscheme which misses the generic fibre has to be &#8220;vertical&#8221;. This intermediate result is interesting by itself.<\/p>\n<p>Does anybody have a reference for this, or similar, results? (I looked in EGA&#8230;)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Yesterday I found this in a preprint by Brian Osserman and Sam Payne: If X &#8212;&gt; S is locally of finite type, and x -&gt; z is a specialization of points in X with z a closed point of its &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=717\">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-717","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\/717","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=717"}],"version-history":[{"count":13,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/717\/revisions"}],"predecessor-version":[{"id":730,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/717\/revisions\/730"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=717"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=717"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=717"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}