PhD COURSE at  ROMA TRE UNIVERSITY
                                                                                                                                        (ACADEMIC YEAR 2014/2015)

 
  Elements of Geometric invariant theory with Applications to Moduli
  by Ian Morrison & Filippo Viviani


NEXT LECTURE: Wednesday 26 November, 17:30--19, Room 311.
Schedule: Monday 15--16:30 (room 009), Wednesday 17:30--19 (room 311).

INTRODUCTION
Geometric invariant theory (or GIT) studies the action of a linear algebraic group G on an algebraic variety X and provides techniques for constructing a quotient of  X by G. 

One of the main applications of GIT is the construction of moduli spaces: in this case, X is usually a parameter space for certain rigidified geometric objects (varieties embedded in projective spaces, sheaves together with a surjection from a given vector bundle , etc..) and the relation of forgetting the extra-stucture is given by an action of a linear algebraic group G.  
 The course will be divided in two parts:
 (1) The first part (by Ian Morrison) will provide an introduction to the theory;
 (2) The secon part (by Filippo Viviani)  will deal with applications to moduli, and in particular the construction of moduli spaces of abelian varieties (and curves).

DURATION
The first part (by Ian Morrison) will consist of 6 hours and it will be given in period 17-28 November 2014.

The second part (by Filippo Viviani) will consist of 8/10 hours and it will be given in the period 1-19 December 2014.



PREREQUISITES:  Basic knowledge of algebraic geometry. 


PROGRAM

PART I (by Ian Morrison)

This part of the course will first explain the construction of the GIT quotient of a variety by a reductive algebraic group. We will work over the field of complex numbers and assume both that the variety acted on is affine and that the group acting is a torus or special linear group. These special cases are far the most common in applications to moduli and, by restricting to them, we can give fairly complete details assuming only familiarity with linear algebra and affine algebraic geometry. However, I will also indicate very briefly what is required to extend the theory to general reductive algebraic groups, to positive characteristic, and to actions on projective varieties, and what pathologies can occur for non-reductive groups.

Then we will study the notions of unstable, semistable and stable orbits which arise naturally out of the geometry of GIT quotients and prove (in the same special cases) the Hilbert-Mumford numerical criterion. This provides an effective tool for identifying these loci that is critical in most applications and that we will illustrate in a variety of examples. Finally, we will discuss the complementary instability theorem of Kempf and its applications.
 

Plan of Lectures:

Actions of linear algebraic groups on affine varieties
Finite generation of rings of invariants
Geometry of quotient maps and of orbit closures
Stable, semistable and unstable loci
The Hilbert-Mumford Numerical Criterion
Kempf's Instabiity Theorem



PART II (by Filippo Viviani)

The second part of the course will deal with the application of GIT to the construction of the moduli space of abelian varieties (and of curves).


Plan of Lectures:

A crash course on abelian varieties over a field
Abelian schemes: duals, polarizations, level structures, deformations
The moduli functor of abelian schemes and the concept of a coarse/fine moduli space
The method of covariants in GIT
The construction of the moduli space of abelian schemes
Application: the construction of the moduli space of curves using the Torelli morphism


REFERENCES

[GIT] D. Mumford, J. Fogarty, F. Kirwan: Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete 34. Springer-Verlag, Berlin, 1994.

[New] P. E. Newstead: Introduction to moduli problems and orbit spaces. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 51.Tata Institute of Fundamental Research, Bombay; Narosha Publishing House, New Delhi, 1978.

[Dol] I. V. Dolgachev: Lectures on invariant theory. London Mathematical Society Lecture Note Series, 296.Cambridge University Press, Cambridge, 2003.

[Muk] S. Mukai: An introduction to invariants and moduli. Cambridge Studies in Advanced Mathematics, 81. Cambridge University Press, Cambridge, 2003.

[Kac] V. Kac: Invariant theory. Notes of a course avalaible at http://people.kth.se/~laksov/notes/invariant.pdf

[Bri] M. Brion: Introduction to actions of algebraic groups. Notes of a minicourse available at  http://www-fourier.ujf-grenoble.fr/~mbrion/notes_luminy.pdf

[Rei] Z. Reichstein: Geometric invariant theory. Notes of a course available at http://www.math.ubc.ca/~maxim/GIT.pdf

[Rei] G. Kempf: Instability in invariant theory. Unpublished notes.

FOR ANY INFORMATIONS OR QUESTIONS: Please contact Filippo Viviani at viviani(at)mat.uniroma3.it