March 2009


So, as of the last post in the series, we defined Schubert cells.  We’re going to use them to discuss the Cohomology of the Grassmannian, and to write down an explicit basis.  With an eye looking forward, next time, we’ll work out the cup product in this cohomology ring, and then finally we’ll use it to solve some problems.  So today, we’ll discuss Cellular Homology, Poincaré Duality, and the cohomology of the Grassmannian.

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A nice thing about elliptic curves is the wealth of information which is tied up in the isomorphism class of the group. Over the complex numbers, every elliptic curve is \mathbf{C}/\Lambda where \Lambda is a lattice of rank 2 contained in the complex numbers. The Mordell-Weil Theorem tells us that when we restrict to a number field we get a finitely generated abelian group.  The Birch and Swinnerton-Dyer conjecture tells us how important the rank is and today we get a glimpse of what the torsion can tell us. The Weil Pairing is a canonical identification of \bigwedge^2 E[m] with the m-th roots of unity (here we work over a field of characteristic not dividing m). (more…)

Hopefully this will be the last background information post before we state and begin the proof of the Riemann Roch Theorem.  This post will be a brief overview of Cycles, Chow Rings, and Chern Classes and their properties.  The briefness is a bit unfortunate since the theory is quite useful and I venture to say all algebraic geometers need to be familiar with it.  Luckily there is a very thorough reference, Fulton’s book on Intersection Theory.  As motivation, remember way back to when you first learned about vector bundles and cohomology in the topological setting;  to a complex vector bundle E \rightarrow B we associate a cohomology groups c_i \in H^{2i}(B, \mathbb{Z}) which in some sense measure how far from trivial the bundle is.  These cohomology classes satisfy some formal properties like functoriality and the whitney sum formula.  One way of constructing these classes is to look at the associated projective bundle P(E) and apply the Leray Hirsch theorem to conclude a certain relationship in the cohomology ring of the bundle.  The beauty is that this can be replicated in algebraic geometry, and it works in a setting a bit finer than cohomology – inside the Chow Ring.  An overview follows…

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It’s been awhile since the last post, but Spring Break happened.  Anyway, back to Schubert Calculus! Last time, we discussed Grassmannians, this time, we’re going to talk about their most important subvarieties, the Schubert Varieties.

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Apologies that I haven’t been posting recently. Sage days was this past weekend along with 6 inches of snow in Georgia(read: power outages), the Arizona Winter school in a week and everything else, it’s been very busy.

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