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!