Math 259 - K3 surfaces
Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu,
Lectures: WF, 10-11:20am, APM 5218.
Office hours: I am
lecture or by appointment. Also, feel free to drop in if you see me in my
The name "K3 surface" was introduced by Andre Weil in honor of three
algebraic geometers, Kummer, Kahler and Kodaira, and the mountain K2 in
course aims to give an introduction to the geometry of K3 surfaces,
focusing on a few selected
topics. In particular, I hope to discuss the following:
- examples of K3 surfaces
- elliptic K3 surfaces
- linear systems on K3s
- Hodge structures
- periods and Torelli
- the moduli space of K3s
- the tautological ring.
(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.
- Lecture 1:
Overview and questions to be addressed in this course. Where K3s stand in
the classification of surfaces.
- Lecture 2: Preliminaries: curves on surfaces, intersection
product, adjunction formula, Riemann-Roch for surfaces. Genus of a K3
surface and connection with genus of curves in the linear series.
- Lecture 3: Construction of K3 surfaces via complete
intersections in projective space. Genera 3, 4, 5. General construction of K3 surfaces via
complete intersections in Fano manifolds of coindex 3.
- Lecture 4: Explicit examples of K3 surfaces and of prime
manifolds of coindex 3. Genera 6, 7, 8, 9, 10, 12. Connection between the
moduli space of low genus K3s and low genus curves.
- Lecture 5: Cyclic covers: formula for
irregularity and for the canonical bundle. Construction of genus 2 K3
surfaces via cyclic covers. Construction of elliptic K3 surfaces as
branched covers of Hirzerbruch surfaces.
- Lecture 6: The topology of K3 surfaces. The Betti/Hodge
numbers, topological Euler characteristic, the signature of the
intersection form, the
second integral cohomology as a lattice. The Hodge index
theorem. The positive cone, nef cone, ample cone. Checking nef and
ample via intersections with (-2) curves.
- Lecture 7: Periods and period domain. Torelli
theorems stated. Outline of the construction of the moduli space
K3s via periods. The method via the Hilbert
- Lecture 8: Very ampleness of 3H if H is ample.
Fujita conjecture. Reider's
method and strategy for the proof. Building rank 2 vector bundles via
extensions. Illustration for curves: Max Noether's theorem for canonical
Serre construction over surfaces.
- Lecture 9: Slope stability of torsion free sheaves over
surfaces. Simple sheaves. Bogomolov inequality. Proof of for K3
- Lecture 10: Crash course in Chern classes. Mumford's proof
of Kodaira-Ramanujam. Proof of
Reider's theorem. Global generation of big and nef linear series on K3s,
and the exceptions for elliptic K3s.
- Lecture 11: Globally generated linear series over
K3s: hyperelliptic and non-hyperlliptic case. Projective normality of
linear systems over nonhyperelliptic K3s.
Lattice polarized K3-surfaces.
the elliptic and hyperlliptic loci in terms of lattice polarizations.
- Lecture 12: Characterization of elliptic fibrations in
terms of divisors with square zero.
- Lecture 13: Kodaira's classification of singular fibers in
an elliptic fibration. Extended Dynkin diagrams.
- Lecture 14: The moduli space of K3 surfaces. Coarse and
fine representability. The moduli stack.
- Lecture 15: Crash course on stacks.
Strategy for constructing the moduli stack of K3s and what needs to be
- Lecture 16: The construction of the moduli stack of
of the moduli stack.
- Lecture 17: Special cycles in the moduli space of K3s. The
Picard group. Comparison between moduli of
curves, abelian varieties and K3s. The Hodge bundle. Lambda and kappa
classes in each of these settings.
- Lecture 18: The tautological rings
of M_g, A_g and K_g. Conjectures/results on the largest dimension of a
complete subvariety in each of the three cases. Discussion of the
tautological ring of M_g. Mumford's relation on M_g.
- Lecture 19: Mumford's relation on A_g and the
calculation of the lambda-ring of A_g. The lambda-ring of
A_g is Gorenstein. The lambda-ring of
K_g and its presentation. The Noether-Lefschetz and kappa rings.