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.

]]>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!

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

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

]]>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 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 . Now there is a lot of structure in the irreducible representations of ! 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 as symmetry about hyperplanes of the form for a root of 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…

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