{"id":397,"date":"2010-05-11T21:24:39","date_gmt":"2010-05-11T21:24:39","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=397"},"modified":"2010-05-11T21:24:39","modified_gmt":"2010-05-11T21:24:39","slug":"unipotent-inertia","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=397","title":{"rendered":"Unipotent inertia"},"content":{"rendered":"

My prediction at the end of the last post was complete nonsense! Here are some examples of actions where the stabilizer jumps in codimension 1:<\/p>\n

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  1. The action G_a^n x P^n —> P^n given by (a_1, …, a_n), (x_0: …: x_n) maps to (x_0: x_1 + a_1x_0: … : x_n + a_nx_0). The generic stabilizer is trivial and over the divisor x_0 = 0 the stabilizer is G_a^n. So the dimension of the stabilizer can jump up arbitrarily high in codimension 1.<\/li>\n
  2. A special case of the example above is the case n = 1 which Jarod Alper pointed out. If y = x_0\/x_1 then the action looks like y maps to y\/(1 + ty) where t is the coordinate on G_a.<\/li>\n
  3. Note that there are many formal actions \\hat{G_a} x \\hat{A^1} —> \\hat{A^1}, because if theta is the derivation ty^k(d\/dy) acting on C[[y]] then if k > 1 we can exponentiate and get automorphisms phi_t = e^theta : C[[y]] —> C[[y]] which satisfy phi_t \\circ phi_s = phi_{s + t}.<\/li>\n
  4. Another example due to Jarod is the action G_a x A^2 —> A^2 given by t, (x, y) maps to (x + ty, y). The locus of points where the stabilizer is G_a is y = 0. This action seems very different from the action in case 2, allthough it may not be so easy to prove.<\/li>\n
  5. Take the product P^1 x P^1 with the action of G_a which is trivial on the first component and as in example 2 on the second. Then we may blow up (several times) in invariant points. If you do this in a suitable manner you will find an exceptional curve E consisting of fixed points where the local ring of the blow up at the generic point of E looks like C[x, y]_{(y)} and where the action is given by (x, y) maps to (x\/(1 + txy^n), y). This gives infinitely many actions which cannot be etale locally isomorphic since the action is trivial modulo y^n and not y^{n + 1}. Note that y is the uniformizer of the local ring in question.<\/li>\n<\/ol>\n

    The conclusion is that if you allow the jump of the inertia group to be non-reductive, then many examples exist (there may even be moduli in the examples).<\/p>\n","protected":false},"excerpt":{"rendered":"

    My prediction at the end of the last post was complete nonsense! Here are some examples of actions where the stabilizer jumps in codimension 1: The action G_a^n x P^n —> P^n given by (a_1, …, a_n), (x_0: …: x_n) … 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-397","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\/397","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=397"}],"version-history":[{"count":3,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/397\/revisions"}],"predecessor-version":[{"id":400,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/397\/revisions\/400"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=397"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=397"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=397"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}