The aim of the course is to study moduli problems in algebraic geometry and the construction of moduli spaces via geometric invariant theory. A moduli problem is a classification problem, where we have a class of objects we want to classify up to some equivalence relation; for example, hypersurfaces in a projective space up to the automorphisms of the projective space or vector bundles on a variety up to isomorphism. A moduli problem is formalised by a moduli functor and a (fine) moduli space is a scheme that represents this functor. Typically moduli spaces are constructed as a quotient of a parameter space by a group of equivalences. The construction of algebraic quotients, as opposed to topological quotients, is given by geometric invariant theory. In the course, we will study moduli functors, algebraic groups and their actions, affine quotients and projective quotients, criteria for semistability, as well as some classical moduli problems (if there is sufficient interest and time permits, we will cover the construction of moduli spaces of vector bundles on a smooth projective curve).

Follow up seminar: Further topics in GIT (such as toric GIT, symplectic quotients and variation of GIT), Summer Semester 2016.

Exam : Tuesday 10-12, 16th February, SR007/008 Arnimallee 6 (written).

Second exam: Friday 15th April, Room 011 Arnimallee 3 (oral); please email me by 11.04.16 to arrange a time.

Lectures : Tuesdays 10-12, SR210 Arnimallee 3.

Exercises : Tuesdays 14-16, SR130 Arnimallee 3. (Hand in: Monday 10am)

**Lecture notes**

Complete Lecture Notes.

Lecture Notes by week:

Lecture 1 (13/10/15): functor of points, moduli problems and moduli functors.

Lecture 2 (20/10/15): fine and coarse moduli spaces, pathologies, constructing moduli spaces.

Lecture 3 (27/10/15): algebraic groups and algebraic actions.

Lecture 4 (03/11/15): categorical quotients, good quotients and geometric quotients.

Lecture 5 (10/11/15): Hilbert's 14th problem I: unipotent and reductive groups.

Lecture 6 (17/11/15): Hilbert's 14th problem II: Weyl's unitary trick, Reynolds operators and Nagata's Theorem.

Lecture 7 (24/11/15): The affine GIT quotient: it is a good quotient and restricts to a geometric quotient on the stable set.

Lecture 8 (01/12/15): The projective GIT quotient: (semi)stability, S-equivalence and polystability.

Lecture 9 (08/12/15): Linearisations: the definition, semistability and GIT quotients.

Lecture 10 (15/12/15): The Hilbert-Mumford criterion for (semi)stability I.

Lecture 11 (05/01/16): The Hilbert-Mumford criterion for (semi)stability II.

Lecture 12 (12/01/16): Moduli of projective hypersurfaces (and plane cubic curves in particular).

Lecture 13 (19/01/16): Moduli of vector bundles on a curve I: The Riemann-Roch formula.

Lecture 14 (26/01/16): Moduli of vector bundles on a curve II: Semistability and boundedness.

Lecture 15 (02/02/16): Moduli of vector bundles on a curve III: The quot scheme and GIT set up.

Lecture 16 (09/02/16): Moduli of vector bundles on a curve IV: The construction.

References

**Exercise sheets**

Exercise Sheet 1.

Exercise Sheet 2.

Exercise Sheet 3.

Exercise Sheet 4.

Exercise Sheet 5.

Exercise Sheet 6. Solution: Ex 2 .

Exercise Sheet 7.

Exercise Sheet 8.

Exercise Sheet 9.

Exercise Sheet 10.

No Sheet 11.

Exercise Sheet 12.

Exercise Sheet 13.

Exercise Sheet 14.

Exercise Sheet 15 (Revision Sheet).

For questions about the exercise sheets, please contact Eva Martínez.