Topics in algebraic geometry: algebraic surfaces

Lectures: 2 sessions / week, 1.5 hours / session

The prerequisite is a first course in algebraic geometry, though the course will be mostly self-contained and we will introduce background material as needed.

This semester, 18.727 will be a graduate course on algebraic surfaces. The main aims will be to go over the classification of surfaces (Engriques-Castelnuovo for characteristic zero, Bombieri-Mumford for characteristic p), while working out plenty of examples, and treating their geometry and arithmetic as far as possible. The exact choice of topics depends on the interests of the students.

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.

Occasional problem sets will be handed out. Graduate students do not have to hand these in for credit, but undergraduates taking the course do. Students are encouraged to work together on the problems.

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