# Math 624/5, Algebraic Geometry, 2008/2009

** Instructor: **
Ching-Li Chai

*Office*: DRL 4N36, Ext. 8-8469.

*Office Hours*: W 11:00-12:00.

Email:`chai@sas.upenn.edu`

#### General Information

**Lectures**: Mondays and Fridays 1:30-3:00, 3C2 DRL
(Spring 2009)

**Topics**:
Definitions of schemes and sheaves, elementary global
properties of Proj(R), Zariski's main theorem,
intersection multiplicities, group schemes, cohomology
of coherent sheaves, Riemann-Roch theorem for curves,
birational geometry of surfaces, GAGA and GFGA,
theory of descent.

**Background knowledge**: A good understanding
of basic algebra at the level of standard first year graduate courses
is required, especially: maximal ideals and prime ideals of
commutative rings, noetherian rings and modules, localization of
rings and modules, projective modules, flatness, completions.
We will explain materials on commutative algebra as needed,
but supplementary readings on commutative algebra may be helpful.

**Text**: We will roughly follow Mumfords's
"Algebraic Geometry II", coauthored with Tadao Oda, for the
theory of schemes, plus materials on the
theory of algebraic curves and surfaces.

Please let me know if you find any typo/error in the manuscript.

Homeworks/Exercises

**Some references:**
- Eisenbud, Commutative Algebra with a View Toward Algeraic Geometry
- Griffiths-Harris, Principles of Algebraic Geometry
(transcendental approach to complex algebraic geometry)
- Harris, Algebraic Geometry
- Hartshorne, Algebraid Geometry
(a textbook that has dominated the market)
- Matsumura, Commutative Algebra
- Matsumura, Commutative Ring Theory (a standard textbook for
commutative algebra)
- Mumford, Algebraic Geometry I, Complex Projective Varieties
- Mumford, Lectures on Curves on Algebraic Surfaces
- Mumford, Abelian Varieties

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