Groups, actions, and Chow groups

It’s been two months since I started blogging, and I am enjoying it immensely. I’ve also given three talks in that time and felt that two of them in particular went extremely well. I have some suspicions that practicing articulation via blogging correlates with delivering good talks, which was one of my initial motivations for starting this up.

Today I gave a talk as part of an intersection theory learning seminar on various appearances of equivariance in Chow groups. I talked about the interaction between root systems and del Pezzo surfaces that has been the subject of recent posts as well as some more technical matters.

One of these was dynamic projection. A strategy in computing the multiplicative structure of Chow rings is to ‘specialise’ to a particularly accessible representative of a rational equivalence class. One way of doing this is to use dynamic projection, which employs the automorphisms of a space to move cycles around. The basic idea is to take an arc of automorphisms \{\psi_t\}_{t\in\mathbb{G}_m} based at \psi_1=\text{id} and make a rational equivalence from a cycle Z\subset X via

W^\circ=\bigcup_{t\in\mathbb{G}_m}\{t\}\times\psi_t(Z)\subset\mathbb{G}_m\times X

to the flat limit as t\to 0 which is the fibre of the closure W\subset\mathbb{P}^1\times X of W^\circ over 0\in\mathbb{P}^1. For many applications – such as calculations in Schubert calculus – dynamic projection in \mathbb{P}^n is enough. Here \text{Aut}(\mathbb{P}^n)=\text{PGL}_{n+1} (when the ground ring is factorial at least…) and so one source of arcs is projective representations of \mathbb{G}_m. This is sufficient to produce many interesting (and fun! and sometimes useful) computations. Indeed, while I was preparing my talk I realised how apt the language of projective representations is to describe dynamic projections of this type in \mathbb{P}^n. As a result, I documented my thoughts along with translations of some of the main results in Eisenbud and Harris’ book on intersection theory in terms of projective representations and several [possibly trivial] questions to which I would be interested to know the answers.

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