Math 206 - Points on surfaces
Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu,
Lectures: WF, 1:30-2:50am, APM 7421.
Office hours: I am
lecture or by appointment. Also, feel free to drop in if you see me in my
The topic for Math 206 this quarter is the Hilbert scheme of
points on surfaces.
In the beginning of the course, we will construct the Hilbert scheme
in full generality. This is important for a number of moduli problems in
For the Hilbert scheme of points on surfaces, we will compute some
of the topological invariants. In particular, we express the Euler
characteristics in terms of modular forms. Connections with the
representation theory of affine Lie algebras will be made.
(This may be a bit too ambitious for a one quarter course, so the
may change slightly as we go.)
- I will make an attempt to be
as self-contained as the topic permits. However, I will assume
algebraic geometry at the level of Math 203.
- The course is intended for graduate
students in Mathematics, though everybody is welcome. There will be no
homework or exams for this
course if you are a math graduate student; the grade will be based
entirely on attendance.
- There will be no lectures on January
25 and 27. Make-up lecture will be scheduled on Monday, February 13 and
Monday, February 27.
- Lecture 1: Introduction to Hilbert schemes of
points and problems to be addressed: the cohomology of the
- Lecture 2: The Quot functor. A short discussion of flatness.
Representability. Strategy for constructing the Hilbert scheme.
- Lecture 3: Results about regularity of sheaves on projective
space. Boundedness of the Quot functor.
- Lecture 4: Embedding of the Quot functor into the
Grassmannian. Construction of the Quot scheme via the flattening
- Lecture 5-6: Proof of the flattening stratification.
Generic flatness. Dimensional reduction via pushforwards.
Finishing the construction of the Quot scheme over arbitrary
- Lecture 7: Projectivity of the Quot scheme. Deformation
theory of Quot schemes. Smoothness of the Hilbert scheme of points.
- Lecture 8-9: Simple examples of Quot and Hilbert schemes.
scheme over P^1 and its topological invariants: Euler characteristics,
generators for cohomology. Equivariant localization. BB stratification.
- Lecture 10: Hilbert scheme of points over affine plane in
terms of quivers and their representations.
- Lecture 11-12: Euler characteristics of Hilbert scheme of
points for all smooth surfaces. Betti numbers of the Hilbert scheme of
points of toric surfaces.
- Lecture 13: Crash course in GIT and some examples.
Hilbert scheme/symmetric product of points on C^2 as GIT quotients.
- Lecture 14: A brief introduction to Nakajima quiver varieties.
- Lecture 15: Holomorphic 2-forms on the Hilbert scheme of
points. The case of points on C^2. Dimension estimates for the punctual
- Lecture 16: Heisenberg and Clifford algebras and their
representations. Motivation and statement for Nakajima's theorem. Review
- Lecture 17-18: Construction of the Nakajima operators. Nakajima basis.
Commutators of Najakima operators. Intersection numbers involving punctual Hilbert
- Lecture 19: Some applications of Hilbert scheme of points
to enumerative geometry. Some conjectures and known results.