Topics in Algebraic Geometry

The class does not follow a single textbook. Here are some background references:

Badescu, Lucian. Algebraic Surfaces . Translated by Vladimir Masek. New York, NY: Springer, 2001. ISBN: 9780387986685.

Barth, Wolf P., Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven. Compact Complex Surfaces . 2nd ed. Berlin, Germany: Springer, 2004. ISBN: 9783540008323.

Beauville, Arnaud. Complex Algebraic Surfaces . 2nd ed. New York, NY: Cambridge University Press, 1996. ISBN: 9780521495103.

Bombieri, E., and D. Mumford. “Enriques’ Classification of Surfaces in Characteristic p, III.” Invent Math 35 (1976): 197-232.

Griffiths, Phillip, and Joseph Harris. “Surfaces.” Chapter 4 in Principles of Algebraic Geometry . New York, NY: Willey-Interscience, 1994. ISBN: 9780471050599.

Hartshorne, Robin. “Surfaces.” Chapter 5 in Algebraic Geometry . New York, NY: Springer, 1997. ISBN: 9780387902449.

Kodaira, K. “On Compact Complex Analytic Surfaces: I.” Ann of Math 71 (1960): 111-152.

———. “On Compact Complex Analytic Surfaces: II.” Ann of Math 77 (1963): 563-626.

———. “On Compact Complex Analytic Surfaces: III.” Ann of Math 78 (1963): 1-40.

Mumford, D. “Enriques’ Classification of Surfaces in Characteristic p, I.” Global Analysis: Papers in Honor of K. Kodaira . Edited by Donald Clayton Spencer and Shokichi Iyanaga. Princeton, NJ: Princeton University Press: 1969, pp. 325-339. ISBN: 9780691080772.

Topics: The Grassmannian: An Example of a Moduli Space · READINGS: 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.

Kleiman, S. L., and Dan Laksov. “Schubert Calculus.” American Mathematical Monthly 79 (1972): 1061-1082.

Topics: The Moduli Space of Curves · READINGS: 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.

Topics: The Cohomology of the Moduli Space of Curves · READINGS: 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.

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

———. “The Picard Groups of the Moduli Spaces of Curves_._” In Topology. Vol. 26, 1987, pp. 153-171.

Topics: The Moduli Space of Curves is of General Type · READINGS: 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.

Topics: The Kontsevich Moduli Space of Stable Maps · READINGS: Fulton and Pandharipande have a great article. This is probably the best introduction to the subject.