{"id":3299,"date":"2013-06-29T20:51:51","date_gmt":"2013-06-29T20:51:51","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3299"},"modified":"2013-06-29T20:51:51","modified_gmt":"2013-06-29T20:51:51","slug":"unobstructed-in-codimension-3","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3299","title":{"rendered":"Unobstructed in codimension 3"},"content":{"rendered":"

So this is a follow up on the post about Burch’s theorem<\/a>. Namely, I’ve just learned in the last month or so that the next case of this is in Eisenbud + Buchsbaum Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3<\/a>. It says that the resolution of a codimension 3 Gorenstein singularity R\/I with R regular has a free resolution of the form<\/p>\n

0 —> R —> R^n —f—> R^n —> R<\/p>\n

where f is an alternating matrix and the other arrows are given by Pfaffians of f.<\/p>\n

Moreover, if R\/J is an almost complete intersection of grade 3, then R\/J is linked to a Gorenstein R\/I as above and a similar type of resolution can be obtained (results of Brown, kustin, etc).<\/p>\n

OK, this is cool, very cool.<\/p>\n

It seems completely clear that similarly to Burch’s theorem this implies that such a singularity is unobstructed, just as in the codimension 2 Cohen-Macaulay case. To be precise, as a simple consequence of the paper we obtain:<\/p>\n

If R = k[[x, y, z]] and R —> S is an Artinian quotient ring such that either (1) S is Gorenstein, or (2) the kernel of R —> S is generated by at most 4 elements, then the miniversal deformation space of S is a power series ring over k.<\/p><\/blockquote>\n

Right…?<\/p>\n

What I’d like is a reference to articles (with page and line numbers) stating exactly<\/strong> the above for (1) and (2). A generic reference to unobstructedness of determinantal singularities doesn’t count. I’ve googled and binged, but no luck so far. Can you help?<\/p>\n

Or maybe this is just one of the innumerable results in our field that are so clearly true that you cannot formulate it in a paper as your paper will be immediately rejected?<\/p>\n","protected":false},"excerpt":{"rendered":"

So this is a follow up on the post about Burch’s theorem. Namely, I’ve just learned in the last month or so that the next case of this is in Eisenbud + Buchsbaum Algebra structures for finite free resolutions, and … 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-3299","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\/3299","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=3299"}],"version-history":[{"count":9,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3299\/revisions"}],"predecessor-version":[{"id":3308,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3299\/revisions\/3308"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3299"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3299"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3299"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}