Notes

I collect here some expository notes.

1. Finiteness of good moduli spaces of graded points. We show that, for a good moduli stack X, quasi-compact connected components of Grad(X) have good moduli spaces that are finite over that of X. As an intermediate step, we establish that the corresponding connected components of Filt(X) have categorical moduli spaces. (pdf)

2. Notes from a talk on the Hilbert-Mumford criterion. A proof of the Hilbert-Mumford criterion in the projective-over-affine setting is sketched. (pdf)

3. Linearly reductive algebraic groups in characteristic 0. An expository account of the equivalence between reductivity and linear reductivity in characteristic 0. It also contains a proof of finite generation of the ring of invariants. (pdf)

4. The absolute Galois group of C(t) (in french). An exposition of a proof that the absolute Galois group of C(t) is isomorphic to the free profinite group on the set of complex numbers. (pdf)