Actions of algebraic groups appear in many problems of Algebraic Geometry, encoding certain symmetries of the problem at hand. Often one is interested in "factoring out" these symmetries and produce a new geometric object, a so-called "quotient" that parametrises the orbits of the action and hence the isomorphism classes of objects at hand.
The aim of the course is to give an introduction to the theory of algebraic groups, to study their actions on algebraic varieties via Invariant Theory, and to investigate under which conditions quotient spaces can be constructed. We will cover both the classical case of actions on affine varieties, e.g. on affine spaces associated with linear representations of the group under discussion, as well as the case of actions on projective varieties. We will also discuss applications to moduli problems, e.g. to the moduli problem for hypersurfaces in projective space. If time permits, towards the end of the course we will describe some transcendental methods (Kempf-Ness theory) used in the investigation of actions of complex reductive groups.
The course should be accessible to anyone with a firm background in the basic aspects of (classical) algebraic geomety, i.e., the definition and basic properties of (affine and quasi-projective) varieties as well as of morphisms between them, cf. Mumford's Red Book of Varieties and Schemes, Shafarevich's Basic Algebraic Geometry I or Chapters I and II of Hartshorne. Scheme theory will not be assumed or used in any serious manner. The specific things used can be found in the appendices of Kraft's book or the first chapter of Humphreys's book (see below).
* Humphreys: Linear Algebraic Groups, Springer, 1975
* Kraft: Geometrische Methoden in der Invariantentheorie, Vieweg, 1985
* Mukai: An Introduction to Invariants and Moduli, Cambridge University Press, 2003
* Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute, 1978
* Wallach: Geometric Invariant Theory, Springer, 2017