{"id":4559,"date":"2020-04-28T20:15:50","date_gmt":"2020-04-28T20:15:50","guid":{"rendered":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4559"},"modified":"2020-04-29T02:27:47","modified_gmt":"2020-04-29T02:27:47","slug":"purity-part-3","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4559","title":{"rendered":"Purity (Part 3)"},"content":{"rendered":"

This is a continuation of the post Purity (part 2)<\/a>. Thanks to an email from János Kollár we now know that the answer to the question is no for relative dimension ≥ 2 as I will explain in this post. All mistakes in this post are mine (of course).<\/p>\n

Take a large integer n. The minimal versal deformation space (Y, y) of the rank 2 locally free module O + O(n) on P^1 has dimension n – 1 and is smooth. Let X’ be the projectivization of the universal deformation over Y x P^1. Then X’ → Y is a smooth projective family of Hirzebruch families. The fibre Σ = X’_y is the Hirzebruch surface Σ → P^1 which has a section σ with self-intersection -n. Recall that the Picard group of Σ is generated by σ and a fibre F. Consider the invertible module L on X’ whose restriction to Σ is D = -K + (n – 2)F. A computation shows that<\/p>\n

    \n
  1. D σ = 0<\/li>\n
  2. H^1(O_Σ(D)) = H^2(O_Σ(D)) = 0<\/li>\n
  3. H^0(O_Σ(D)) has dimension 3n + 3 and gives an embedding of the contraction of σ in Σ into P^{3n + 2}<\/li>\n<\/ol>\n

    However, every other fibre of X’ → Y is a Hirzebruch surface whose directrix has self-intersection > -n. Hence -K + (n – 2)F will not contract the directrix on any other fibre. We conclude that L defines a factorization X’ → X → Y where we are constracting σ on Σ to a point x in X and nothing else. Thus f : X → Y is a morphism of varieties, Y is smooth, X is a normal variety, and f is smooth at all points except at x. Thus we see<\/p>\n

    codim Sing(f) = n – 1 + 2 = n + 1<\/p>\n

    Since the question was whether codim Sing(f) ≤ 1 + 2 we see the answer is very much no in the case of relative dimension 2.<\/p>\n

    For relative dimension d ≥ 2 we take the morphism X x A^{d – 2} → Y which has a singular locus still of codimension n + 1 and hence this shows the answer is no as soon as n > d.<\/p>\n

    As far as I know the question remains unanswered for relative dimension 1 (besides the one subcase of relative discussed in the previous post on this topic). Please let me know if you have an idea or an example.<\/p>\n

    PS: As János points out the morphism f constructed above is even flat and the fibres of f have rational singularities. Thus it seems unlikely there is a class of singularities strictly bigger than the lci ones (see previous post) for which purity holds.<\/p>\n

    PPS: Another question is whether examples like this can help us find examples of other purity statements gone wrong.<\/p>\n","protected":false},"excerpt":{"rendered":"

    This is a continuation of the post Purity (part 2). Thanks to an email from János Kollár we now know that the answer to the question is no for relative dimension ≥ 2 as I will explain in this post. … 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-4559","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\/4559","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=4559"}],"version-history":[{"count":10,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4559\/revisions"}],"predecessor-version":[{"id":4571,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4559\/revisions\/4571"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4559"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4559"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4559"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}