## 主题目录

### Course Description

The topics for this course vary each semester. This semester, the course aims to introduce techniques for studying intersection theory on moduli spaces. In particular, it covers the geometry of homogeneous varieties, the Deligne-Mumford moduli spaces of stable curves and the Kontsevich moduli spaces of stable maps using intersection theory.

• ## Prerequisites

Algebraic Geometry (18.725). This is a first year graduate class in algebraic geometry at the level of the second and third chapters of R. Hartshorne (Algebraic geometry. New York, NY: Springer-Verlag, 1977). Familiarity with algebraic topology helpful.

• ### 教学讲稿(Lecture Notes)

These notes are somewhat rough: they may contain some mistakes, so please use caution when reading them.

Some notes about the construction of the Hilbert scheme. Included are some notes on G.I.T. and the construction of the moduli spaces.

Some notes containing a brief survey of the cohomology of the moduli space of curves and the Harer-Zagier formula for the orbifold Euler characteristic. I added a few words outlining other work of Harer on the stability of the cohomology of the moduli space of curves.

Some notes describing the Picard group of the moduli space of curves.

Some notes about the Kodaira dimension of the moduli space of curves.  You should also read Chapter 5 and 6F of  Harris, J., and I. Morrison. Moduli of Curves. New York, NY: Springer-Verlag, 1998, p. 366. ISBN: 0387984380. (U.S.)

Some notes on the Kontsevich moduli space of genus zero stable maps.

• ### 其他教学资源

The following table lists suggested readings.

The Grassmannian: An Example of a Moduli Space  Griffiths and Harris. Principles of Algebraic Geometry. New York, NY: Wiley, 1994, c1978, pp. 193-211. ISBN: 0471050598.

Fulton. Intersection Theory. Berlin, Germany; New York, NY: Springer, c1998, chapter 14. ISBN: 354062046X.

Harris. Algebraic Geometry: A First Course. New York, NY: Springer-Verlag, 1992, lectures 6 and 16. ISBN: 0387977163.

Vakil, R. A Geometric Littlewood - Richardson Rule. (PDF)

The following is a very pleasant survey article:

Kleiman, S. L., and Dan Laksov. "Schubert Calculus." American Mathematical Monthly 79 (1972): 1061-1082.
The Moduli Space of Curves You can find a sketch of the G.I.T. construction in:  Harris and Morrison. Moduli of Curves. New York, NY: Springer-Verlag, chapter 4. ISBN: 0387984291.

For the construction of the Hilbert scheme look at:
- Mumford. Lectures on Curves on an Algebraic Surface. Princeton, NJ: Princeton University Press, 1966.
- Sernesi. Topics on Families of projective schemes. Kingston, ON: Queen's University, 1986.
- Grothendieck's original lectures given in the Seminaire Bourbaki.
Grothendieck, Alexander. Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert. (French) (Construction techniques and existence theorems in algebraic geometry. IV. Hilbert schemes). Séminaire Bourbaki. Vol. 6, Exp. No. 221, 249-276.

For a very detailed treatment of G.I.T. look at:  Mumford. Geometric Invariant Theory. Berlin, Germany; New York, NY: Springer-Verlag, 1994. ISBN: 3540569634 (Berlin), and  0387569634 (New York). (There are newer editions with appendices by Fogarty and Kirwan.)

For a detailed proof of the Potential Stability Theorem see:  Gieseker. Lectures on Moduli of Curves. Berlin, Germany; New York, NY: Springer-Verlag, 1982. ISBN: 0387119531. (U.S.)

A good account of the G.I.T. construction is contained in: Mumford, D. "Stability of projective varieties." Enseignement Math 23, no. 2 (1977): 39-110.
The Cohomology of the Moduli Space of Curves A great source is: Harer, J. "The cohomology of the moduli space of curves, in Theory of Moduli." Lectures given at C.I.M.E. Springer Lecture Notes in Mathematics. Vol. 1337. Springer Verlag, Berlin, 1988.

Check out this paper by Hain and Looijenga, it contains a survey of the stuff we have been talking about. (PS)

Arbarello, E., and M. Cornalba. "Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves." J Alg Geom 5 (1996): 705-749.

———. "Calculating cohomology groups of moduli spaces of curves via algebraic geometry." Mathématiques de l'IHÉS 88 (1998): 97-127. (PDF)

———. "The Picard Groups of the Moduli Spaces of Curves." In Topology. Vol. 26, 1987, pp. 153-171.
The Moduli Space of Curves is of General Type The following papers of Joe (with Mumford and Eisenbud) developed the theory:

Harris, J., and D. Mumford. "On the Kodaira dimension of the moduli space of curves." Invent Math 67, no. 1 (1982): 23-86. (With an appendix by William Fulton.)

Harris, J. "On the Kodaira dimension of the moduli space of curves II: The even-genus case." Invent Math 75, no. 3 (1984): 437-466.

Eisenbud, D., and J. Harris. "The Kodaira dimension of the moduli space of curves of genus." Invent Math 90, no. 2 (1987): 359-387.
The Kontsevich Moduli Space of Stable Maps Fulton and Pandharipande have a great article. This is probably the best introduction to the subject.

Fulton and Pandharipande. "Notes on stable maps and quantum cohomology." 1996.

Other papers of interest are:

Pandharipande, Rahul. "Intersections of Q-Divisors on Kontsevich's Moduli Space $\bar{M}_{0,n}(P^r,d)$ and Enumerative Geometry." 1995.

Vakil, Ravi. "The enumerative geometry of rational and elliptic curves in projective space." J Reine Angew Math 529 (2000): 101-153. (Crelle's journal.) (PDF)