November 2010


Let {\pi:\tilde{C}\rightarrow C} be an unramified double cover, where {C} is geneus {g}. Then {\tilde{C}} has genus {2g-1} by the Riemann-Hurwitz formula. Now, {J(C)} encodes lots of information about the geometry of {C}, especially with the additional data of the theta divisor. It turns out that for double covers, there’s an abelian variety that contains a lot of this data.

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Today, we’re going to talk about a very important class of rational varieties, that show up all the time, in quite a variety of different contexts, and at the end, we’ll talk about why.

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Let {M=\mathbb{P}^{nm-1}} be the projectivization of {\mathrm{Mat}_{n\times m}(\mathbb{C})}. Then for all {k}, we have a variety {M_k} given as the set of matrices of rank at most {k}, which is given by the vanishing of the determinents of {(k+1)\times (k+1)} minors. We call these the generic determinantal varieties.

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I know I still have a few readers out there, and some of you are probably technophiles, so here’s a question: I’ve been considering switching from live-texing of things to a digital pen of some sort.  Anyone have any experience with these things? Know which ones work well and which don’t? Which ones may work with linux (though I do have a windows partition if really necessary)?

I’ve looked at a couple of them, and I’m really not sure what their specific pros and cons may be, and I have no real way to try them out directly at the moment.  There’s the LiveScribe pens, which require their magic dot paper stuff, which is a drawback that I’m not sure how annoying it would be, plus I don’t know how useful sound recording would be, though I can see myself maybe using it at some point.  The other brand I’ve looked at is SolidTek’s DigiMemo, which is a bit bulkier being a clipboard, and amazon reviews suggest it’s finicky.  Is there another brand I should look at? I’m mostly looking for a way to nicely digitally archive all of my scratchwork (my collection of notebooks is expanding too fast!) and also maybe taking notes at seminars, conferences and the occasional advanced course.

So, anyone? I figure that any math person who has a digital pen or has at least considered them will have a lot of the same uses in mind as me, and so I’m very interested in opinions.

Pretty much everything in this post is in Mumford’s “Curves and their Jacobians,” but I do a couple of things slightly differently, and I intend to supply a bit more detail in some places.  The goal here is to construct the moduli of curves of genus g for small g.

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Today, we’re going to do something completely different, but which most of my peers seem not to have seen, but is a very cool application of algebraic geometry.

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Today’s post will be, as the title says, a bit short.  It will, more-or-less finish our current discussion of theta characteristics, and then we’ll get back to something else.  But we’ll derive a nice case of a classical formula.

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Last time we defined theta characteristics as square roots of the canonical bundle.  Today, we’re going to analyze the notion a bit, and relate them to quadrics in characteristic two.

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For this whole post, we’ll take C to be a curve and J=J(C) the Jacobian of the curve.  We’re going to construct several special subvarieties (not special in any technical sense, though) of J, which encode a great deal of geometric information about C.

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As promised in the last post, I’m making another go at MaBloWriMo…maybe others will as well.  I don’t know if I’m going to have a coherent topic over the course of this month, but I’ll be starting with abelian varieties associated to curves, so expect me to talk about generalizations of Prym varieties eventually (unless I get distracted by something else along the way).  Today, the basic case: Jacobians!

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