PhD COURSE / CORSO DI DOTTORATO
                                                                                                           (ANNO ACCADEMICO  2010/2011)

Geometric invariant theory
Docente: Filippo Viviani

       
Diary of lectures:
  • 11/10/2010: Moduli spaces (good, coarse, fine) and their construction via quotients.
  • 19/10/2010: Generalities on algebraic groups. The problem of finite generation of the ring of invariants. Review of the structure theorems of linear algebraic groups: tori, unipotent, solvable, semisimple and reductive groups.
  • 02/11/2010: Nagata's theorem: finite generation of the ring of invariants for a rational action of a geometrically reductive group on a finitely generated algebra.
  • 11/11/2010: Generalities on group actions: orbits, stabilizers. Quotients: categorical, orbit space, good and geometric. Existence of the good quotient for the action of a reductive group on an affine variety.
  • 16/11/2010: From good quotients to geometric quotients: polystable, regular and stable points. Example: square matrices up to conjugation.
  • 19/11/2010: Equivariant Picard group. Linearization of actions.
  • 29/11/2010: GIT quotients: semistable points.
  • 01/12/2010: Projective quotients. Example: action of the multiplicative group on an affine variety.
  • 06/12/2010: Characterization of semistable and stable points via the affine cone.
  • 07/12/2010: Hilbert-Mumford numerical criterion for semistability and stability.
  • 14/12/2010: Examples: binary forms, quadrics, non singular hypersurfaces, plane cubics.
  • 15/12/2010: Quotients by finite groups: existence for quasi-projective varieties, Hironaka's counterexample. Difference between categorical/good quotients and orbit spaces/geometric quotients.
  • 19/06/2011: The construction of the moduli space of smooth curves via GIT (seminar by Fabio Felici).
 
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 modern theory of GIT was developed by David Mumford in the 60's, using ideas of Hilbert  in classical invariant theory.
Recently, there has been a revitalization of interests in GIT, mainly in connection with:
  • Variation of GIT: flips and wall crossing phenomena (see [DH] and [Tha]);
  • Mori dream spaces (see [HK] and [McK]);
  • (Log-)minimal model of moduli spaces (see [Hye]).

In the first part of the course, I will  present the basic theorems of GIT,  illustrating them in several classical examples.
In the second part of the course, I will deal with some of the above mentioned recent developments of GIT,
in accordance with the interests of the audience.
Program
A temptative program of the first part of the course is the following:
  • Introduction to Moduli Spaces (good, coarse, fine) and construction via quotients. Examples.
  • Definition of quotients: categorical, geometric, good.
  • Finite generation of invariants for a reductive linear algebraic group. Nagata's counterexample to Hilbert's 14th problem for non-reductive groups.
  • Affine quotients.
  • Linearization of actions.
  • Projective quotients.
  • Numerical criteria for (semi)-stability.
  • Local structure theorems: Luna's étale slice theorem;  Hochster-Roberts's theorem on CM.
The above results will be illustrated in the following classical examples:
  • Binary forms;
  • Points in projective spaces;
  • Hypersurfaces;
  • Linear subspaces of projective spaces.
  • Space curves.
Ideally, we could have parallel seminars (run, for example, by students) on some classical constructions of moduli spaces via GIT:
  • Construction of parameter spaces: Hilbert  and Chow schemes; Quot schemes.
  • Moduli spaces of smooth curves and stable curves.
  • Moduli spaces of abelian varieties.
  • Moduli spaces of sheaves.
  • Universal Picard variety and universal vector bundle.
  • Moduli spaces of stable maps.
Prerequisites:  Basic knowledge of algebraic geometry. Some knowledge of linear algebraic groups may be useful, but I will
quickly review all the needed facts.
Basic 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.

[PV] V. L. Popov, E. B. Vinberg: Invariant Theory, in  Algebraic Geometry IV, Encycl. Math. Sci., Vol. 55, Springer-Verlag , 1994.
Advanced References:
[Tha] M. Thaddeus: Geometric invariant theory and flips. J. American Math. Soc. 9 (1996), 691--723.

[DH] I. V. Dolgachev, Y. Hu: Variation of geometric invariant theory quotients. With an appendix by Nicolas Ressayre.Inst. Hautes Etudes Sci. Publ. Math. 87 (1998), 5--56.

[HK] Y. Hu, S. Keel: Mori dream spaces and GIT. Dedicated to William Fulton on the occasion of his 60th birthday. Michigan Math. J. 48 (2000), 331-348.

[McK] J. McKernan: Mori dream spaces.  Japanes Journal Math. 5 (2010), 127--151.

 [Hye] D. Hyeon: An outline of the log minimal model program for the moduli space of curves.
Preprint available at arXiv: 1006.1094.


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