Projective spaces are the most basic algebraic varieties we know (at least in some sense) and rational varieties are those that are as close as possible to being projective spaces. Informally, a rational variety is one admitting a parametrization by projective space. The project of determining which varieties are rational (or not) has led to the development of a large amount of rich and beautiful theory – and many (seemingly straightforward) questions are still open. On a historical note, the rationality probelm for quadric and cubic surfaces was settled “classically”, that is, over one hundred years ago. The rationality of quadric hypersurfaces in general is also relatively easy to deal with, and occupies the bulk of this post. It wasn’t until the 1970s that Clemens and Griffiths proved that smooth cubic threefolds are not rational by identifying a cohomological obstruction – a method involving the intermediate Jacobian. Showing that a given variety is not rational is usually quite difficult; as an example, the problem of determining whether all smooth cubic fourfolds are rational or not is (I believe) still open. There has also been much recent work in identifying other classes of varieties which are “close” to being rational – but that discussion will have to wait for another day. The goal of this post will be more modest, I’ll discuss the definitions and some more basic examples of rational varieties. (more…)
September 27, 2008
September 23, 2008
Ok, rumor has it that Hironaka is claiming a proof of resolution of singularities in positive characteristic. Anyone know anything more about this? Do we have any readers at Harvard that can confirm this rumor or squash it? Do I need to haul myself to Boston on the double? Please, news!
Source: Not Even Wrong
September 16, 2008
Ok, so I’m going to mostly be posting on toric geometry for the near future, and in particular, working out what we need to do some mirror symmetry with it. I’ll be following various things by David Cox, so check out his webpage for more info. Anyway, this post is mostly going to be background on fans, polytopes, and cones so that we can do toric geometry properly next time. In fact, varieties will not be mentioned.
September 9, 2008
Unless there is some specific interest, I think this will be the final post in the series about taking quotients of varieties by actions of group schemes. Recall that everything is being done over an algebraically closed field, . I’ll discuss what happens in the projective case in this post. To cover such a topic completely is of course impossible, but hopefully this post will illlustrate some of the major ideas and results. There are also some more or less completely worked through, hands on, examples to show this subject is still rooted in polynomial computations!
September 8, 2008
And the results are in…I got out of the exam less than a half hour ago. I passed! Whee, I’m now officially a PhD Candidate. I should be able to resume posting next week, once I’ve finished recovering and done some reading.
September 1, 2008
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Finally we’re ready to discuss what sorts of quotients exist when group schemes act on (some) other schemes. Recall that for simplicity, every scheme/variety/morphism in sight is assumed to be over the spectrum of a fixed algebraically closed field, . For further simplicity, assume all schemes are of finite type. In this post, we’ll begin the discussion with the simplest case, that is, the main results involving taking quotients of reductive group schemes acting on affine schemes. This will be the main motivation for generalizing results to the non-affine case which we also begin to do here. This theory is well developed and sometimes a bit technical though I’ve tried to avoid anything too difficult below and hopefully the examples illustrate some of what’s going on. (The main reference is Mumford’s GIT, and proofs of unproven facts can be found there.)