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.