How to construct quotients in algebraic geometry?

par Eliot Kahan

lundi 10 nov. 2025, 10:30 → 11:25 Europe/Paris

The question of defining a quotient object arises naturally in many mathematical contexts.
In differential geometry, we consider a manifold X equipped with a differentiable action of a Lie group G. When the action is free and proper, the set of orbits X/G naturally inherits a differentiable manifold structure.
On the other hand, if the action is no longer free, for example when certain points have finite stabilisers, the quotient ceases to be a manifold and leads to the more general notion of an orbifold.
In algebraic geometry, the same problem arises: how to construct a good quotient of an algebraic variety X under the action of an algebraic group G? We will explain what a ‘good quotient’ means. The geometric invariant theory (GIT) developed by Mumford that provides a framework for defining such quotients while remaining in the category of algebraic varieties (affine or projective).
In the affine case, we can always construct such a quotient X//G, but in the projective case, the situation is more subtle: the quotient does not always exist globally, and we generally have to exclude a subset of points, the so-called unstable points, in order to obtain a well-defined quotient.
This talk will be the opportunity to do an introduction of affine and projective algebraic geometry, as well as to reductive groups, which will be illustrated with as many examples as possible.