Spring 2019

Classes: Tuesdays and Thursdays, 10:00AM--11:15AM in LGRT 1114.

Office hours: Tuesdays 1:00PM--2:00PM and Wednesdays 1:00PM--2:00PM in my office LGRT 1235H.

Homeworks will be posted here and due every two weeks.

HW1: Due Thursday 2/21. pdf. Reference [HKLR87] pdf.

HW2: Due Thursday 3/7. pdf.

HW3: Due Thursday 3/28. pdf.

HW4: Due Thursday 4/18. pdf.

This course is a second course in algebraic geometry. The prerequisites are a basic knowledge of algebraic geometry and commutative algebra. Some prior experience with sheaf cohomology would be useful but not essential. We will study complex algebraic surfaces, introducing the main computational tools of algebraic geometry along the way. Complex surfaces play a central role in algebraic, differential, and symplectic geometry, and number theory.

Tentative syllabus:

0. Topology of algebraic surfaces. Hodge theory.

1. Sheaves and cohomology.

2. Line bundles, divisors, morphisms to projective space.

3. Birational geometry of surfaces.

4. Characterization of the projective plane. del Pezzo surfaces.

5. Ruled surfaces.

6. K3 surfaces.

7. Elliptic fibrations.

8. Surfaces of general type.

Principles of Algebraic Geometry, P. Griffiths and J. Harris, Wiley 2011. googlebooks.

Compact Complex Surfaces, W. Barth, K. Hulek, Chris Peters, and A.van de Ven, Springer, 2003. googlebooks.

Complex Algebraic Surfaces, A. Beauville, Cambridge University Press, 1996. googlebooks.

Algebraic Surfaces and Holomorphic Vector Bundles, R. Friedman, Springer 1998. googlebooks.

Chapters on Algebraic surfaces, M. Reid, 1996. arXiv.