{"id":1010,"date":"2010-11-14T19:56:06","date_gmt":"2010-11-14T19:56:06","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1010"},"modified":"2010-11-17T14:13:27","modified_gmt":"2010-11-17T14:13:27","slug":"more-projects","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1010","title":{"rendered":"More projects"},"content":{"rendered":"<p>Here is a list of projects that make sense as parts of the stacks project. (For a list of algebra projects, see <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=965\">this post<\/a>.)  This list is a bit random, and I will edit it every now and then to add more items. Hopefully I\u2019ll be able to take some off the list every now and then also. If you are interested in helping out with any of these, then it may be a good idea to email me so we can coordinate. It is not necessary that the first draft be complete, just having some kind of text with a few definitions, some lemmas, etc is already a good thing to have. Moreover, we can have several chapters about the same topic, of different levels of generality (the reason this works well is that we can use references to the same foundational material in both, so the amount of duplicated material can be limited).<\/p>\n<ul>\n<li>If X is a separated scheme of finite type over a field k and dim(X) \u2264 1 then X has an ample invertible sheaf, i.e., X is quasi-projective over k.<\/li>\n<li>If f : X &#8212;&gt; S is a proper morphism of finite presentation all of whose fibres have dimension \u2264 1, then etale locally on S the morphism f is quasi-projective. This also works for morphisms of algebraic spaces.<\/li>\n<li>Local duality; see also the corresponding algebra project.<\/li>\n<li>Cheap relative duality for projective morphisms. Start with P^n over a (Noetherian) ring and deduce as much as possible from that.<\/li>\n<li>More on divisors and invertible sheaves, Picard groups, etc.<\/li>\n<li>Serre duality on projective varieties.<\/li>\n<li>Classification of curves.<\/li>\n<li>Quot and Hilbert schemes.<\/li>\n<li>Linear algebraic groups.<\/li>\n<li>Geometric invariant theory. I think that a rearrangement of the material in the first few chapters of Mumford&#8217;s book might be helpful. In particular some of the material is very general, but other parts do not work in the same generality. Note that we already have the start of a chapter discussing the myriad possible notions of a quotient, see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/groupoids-quotients.pdf\">groupoids-quotients.pdf<\/a>.<\/li>\n<li>Resolution of two dimensional schemes.<\/li>\n<li>Semi-stable reduction theorem for curves. (Is there any way to do this without using resolution of singularities of two dimensional schemes or geometric invariant theory?)<\/li>\n<li>Abstract deformation theory a la Schlessinger (but maybe with a bit of groupoids thrown in).<\/li>\n<li>Deformation theory applied to specific cases: zero-dimensional schemes, singularities, curves, abelian varieties, polarized projective varieties, coherent sheaves on schemes, objects in the derived category, etc.<\/li>\n<li>Brauer groups of schemes.<\/li>\n<li>The stack of curves and pointed curves, including Kontsevich moduli stacks in positive characteristic are algebraic stacks.<\/li>\n<li>The stack of polarized projective varieties is an algebraic stack.<\/li>\n<li>The moduli stack of polarized abelian schemes is an algebraic stack.<\/li>\n<li>The stacks of polarized K3 surfaces.<\/li>\n<li>Alterations and smoothness (as an application of moduli stacks of curves above).<\/li>\n<li>Add more here as needed.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Here is a list of projects that make sense as parts of the stacks project. (For a list of algebra projects, see this post.) This list is a bit random, and I will edit it every now and then to &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1010\">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-1010","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\/1010","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=1010"}],"version-history":[{"count":10,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1010\/revisions"}],"predecessor-version":[{"id":1019,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1010\/revisions\/1019"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1010"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1010"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1010"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}