Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu, AP&M 6-101.

Lectures: WF, 1:30-2:50am, APM 7421.

Course description:

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 algebraic geometry.

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 goals may change slightly as we go.)

Prerequisites:

I will make an attempt to be as self-contained as the topic permits. However, I will assume background in 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.

Announcements:

• 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 Summaries

• Lecture 1: Introduction to Hilbert schemes of points and problems to be addressed: the cohomology of the Hilbert scheme.
• Lecture 2: The Quot functor. A short discussion of flatness. Representability. Strategy for constructing the Hilbert scheme. Castelnuovo-Mumford regularity.
• 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 stratification.
• Lecture 5-6: Proof of the flattening stratification. Generic flatness. Dimensional reduction via pushforwards. Finishing the construction of the Quot scheme over arbitrary schemes.
• 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. Quot 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 Hilbert scheme.
• Lecture 16: Heisenberg and Clifford algebras and their representations. Motivation and statement for Nakajima's theorem. Review of correspondences.
• Lecture 17-18: Construction of the Nakajima operators. Nakajima basis. Commutators of Najakima operators. Intersection numbers involving punctual Hilbert schemes.
• Lecture 19: Some applications of Hilbert scheme of points to enumerative geometry. Some conjectures and known results.