|Structural Stable Configurations of Lines of Curvature and Umbilic Points of Surfaces: Jorge Sotomayor y Carlos Gutierrez
At the same time, with the formalization of algebraic geometry, a new branch of Lie theory was created, the theory
of Algebraic Groups (Borel, Chevalley, etc.) finally many mathematicians worked on moduli problems having in mind
the prototype theory of moduli (i.e. parameters) for Riemann surfaces.
One of the fruitful ideas, for classification problems of algebraic objects has been the use of the Invariant
Theory of Hilbert with the necessary updates, this is now called the Theory of Hilbert Mumford (cf. Mumford, Fogarty).
The main point of this Theory is to discuss under which conditions we can construct complete (or projective)
varieties parametrizing orbits under a group action.
In this Theory the main ideas are the description of stable and semistable points (points where not all
invariants without constant term vanish) by the Hilbert Mumford criterion.
In fact the idea is to use Hilbert Theory for the affine cone of a projective variety and then use the
homogeneous invariants to give projective coordinates for some quotient whose points parametrize the closed orbits
in the set of semistable points.
These ideas have led to several distinct paths of research. First, in the theory of moduli the question is
to describe as best as possible which are the stable and semistable points of the problem. Second one may study by
itself the geometry of the quotient, here the main tool is the ètale slice theorem of Luna which reduces the
study of the singularity of a point in the quotient to the analogous problem for the image of the origin in the normal
representation to the closed orbit corresponding to the given point.
A further topic comes from the connections with symplectic geometry and is the Kempf-Ness theory and that of
Kirwan, here the moment map plays a major role. This theory is also suitable for the study of quotients under compact
group actions (in contrast to the algebraic groups of the previous theory).
Finally the Russian school (Popov, Vinberg and others) have investigated the problem of classifying the good
representations, i.e. the ones for which the quotient map has particularly good behaviour (for instance the quotient is
smooth or the map is flat, equidimensional etc.).