Canonical curves and the Eagon-Northcott complex

Happy thanksgiving! I’m giving a talk in the student commutative algebra seminar next week on canonical curves and Green’s conjecture. When I first saw it, it seemed like another one of those algebraic conjectures where two niche invariants coincide. But it turns out that there’s actually a fair amount of geometry at work behind the scenes.

The base case of Green’s conjecture – for Veronese curves or, equivalently, canonical models of hyperelliptic curves – is precisely the case that the Eagon-Northcott complex is designed to cater for. Here are my notes supporting the brief allusion to it that I’ll make during my talk.

And my actual talk notes are available here. Canonical rings are naturally motivated for a geometer, who automatically cares about the MMP, and it’s nice to engage more with their underlying commutative algebra to see what geometry is reflected in algebra that’s slightly away from the beaten path.

The geometry of continued fractions

I’m enjoying my post-qual liberty – I passed! – by giving a talk later today for Women in Math at Berkeley on continued fractions, and the elegant efficiency they offer for performing computations in the geometry of surfaces. Here are my notes.

I suspect that, quite rightly, many of the talks in an expository forum like this are what my undergraduate advisor would call ‘propaganda talks’: a rough sketch of some of the main ideas and motivations of a field or topic, probably accompanied by some central examples.

Partly out of philosophy, partly out of curiosity, and partly because I tied myself to it in my title and abstract, I’m not going to buck this trend. Instead, I’m going to try to communicate something of the everyday activity of a geometer (at least, in my experience of course). Some big ideas will naturally be introduced into the story as required – toric geometry, intersection theory, resolution of singularities,… – but I’m hoping to drape the talk on the many beautiful and elementary examples of things I’d like to compute (for instance, invariant monomials under a group action). I anticipate that it won’t be the easiest talk to deliver or receive, but I’m intrigued to see what happens!

Blowups in detail

Today I reconciled the three main presentations of blowups that I know of. Again, it’s fun comparing the classical situation of primarily blowing up points (on surfaces, say) with some of the more disparate things that can happen (or not happen) when blowing up higher dimensional subvarieties. These notes prioritise conciseness and nonstandard examples…

Selected qual(ity) examples

I thought I might post a list of some examples that have cropped up during qual preparation and especially during the mock quals I’ve had so far. Thanks to the people who’ve helped me to see these examples in the nicely presented form in which they appear here. It wouldn’t be outrageous to expect that I’ll add to this list over time.

Riemann theorems

Today was the day I recapped some of the foundational theorems in algebraic geometry in preparation for my qual. It struck me how infrequently I usually work with curves and their massive divisor groups – I cut my teeth on intersection theory of surfaces – so I took some pleasure in reviewing some classical facts about line bundles on curves from my qual reference Gathmann, and from Vakil’s class notes. There are some goodies about hyperelliptic curves of genus 2 and 3 to keep you (or at least me) entertained.

From Lie algebras to Lie groups

I was given a mock qual for representation theory yesterday during which some content naturally cropped up which was just off the borders of my formal syllabus for Lie theory. I thought it might be a good idea to cover this in some more detail, as well as review the Lie group theory that actually does appear on my syllabus. I’m getting a lot out of writing these summaries; communication is good!

Toric varieties as quotients: changing the status quo-tient

I’ve had a nice Saturday afternoon brushing up on something else toric that I half-learned once. Briefly, instead of the local + gluing construction of normal toric varieties from fans, one can construct them holistically as quotients by some reductive group (either a torus, or a product of a torus and a finite group). This global construction has certain benefits: one builds a ‘total coordinate ring’ that plays the same role as the (graded) polynomial ring does for projective space in allowing easy access to closed subvarieties (as homogeneous ideals) and quasicoherent sheaves (as graded modules).

Indeed, the grading here is by the class group, which for (weighted) projective spaces is just the familiar $\mathbb{Z}$-grading. Notice also that the construction doesn’t require the toric varieties produced to be projective.

Representations of Lie algebras

…and here is the second half of the representation theory on my qual: representations of Lie algebras. Again, it’s intentionally – or reassuringly – concise (for example, I suppress the Killing form entirely).

Representations of quivers

It’s been quite some time since I last blogged mostly due to qual-related things. As I enter the last stretch of preparation, I’m currently making some intense synopses charting out the main narratives of each of my topics. Half of the representation theory portion is on representations of quivers culminating in Gabriel’s theorem, which is a sumptuous story to tell for its own sake.

I thought I’d post my synopsis here for reasons of katharsis and, hopefully, for your viewing pleasure. Expect other similar posts soon.

Weyl invariance

I haven’t blogged for a while due to various occupations of my time – I returned to America, moved into a new place, started the school year,… – however I plan to do so on a reasonably regular basis now that my routine is up and running.

I have my qual coming up in October and so most of my time is spent preparing for that at the moment. One of my topics is the representation theory of Lie groups and algebras. I was taught a course on Lie algebras by a finite group theorist who refused to cover any more representation theory than was necessary to classify the semisimple complex Lie algebras, and so my last week or so has been spent extending my rigorous knowledge of root decompositions to a similarly firm grasp of my intuition said had to be true for arbitrary representations of semisimple Lie algebras.

One of the aspects of the theory that I had underappreciated at the first time of learning was why one should expect Lie theoretic objects to be invariant under the action of the associated Weyl group. As my notes describe, the appearance of the Weyl group is totally unsurprising when considered from the right point of view.

The roots of a (semi)simple Lie algebra $\mathfrak{g}$ come in +/- pairs and together produce a one dimensional subspace of the designated Cartan subalgebra as their commutator. The sum of these three 1-dimensional spaces is a subalgebra isomorphic to $\mathfrak{sl}_2(\mathbb{C})$. Now there is a lot of structure in the irreducible representations of $\mathfrak{sl}_2(\mathbb{C})$! In particular, their weights or eigenvalues are symmetric about zero and form a connected string. The symmetry manifests in the weights of a representation of $\mathfrak{g}$ as symmetry about hyperplanes of the form $\alpha^\perp$ for $\alpha$ a root of $\mathfrak{g}$ and the orthogonal taken with respect to the Killing form. In other words, reflections in these hyperplanes preserve weights of representations. What is the Weyl group? Exactly the group generated by these reflections.

This whole stream of theory has the effect of reducing many decomposition questions about representations of semisimple Lie algebras to combinatorial activities decomposing very symmetric polygons into other very symmetric polygons. I’ll describe some examples next time…