Topics in Algebraic Geometry I

Instructor: Jungkai Alfred Chen

Office: Old Math. Bldg 108

I. Contents：

1. Algebraic Curves and Algebraic Surfaces

a. Divisors, line bundles

b. Riemann-Roch theorem

c. Cohomology theory

d. Intersection theory, Hodge index theorem

e. Surface singularities

f. Birational maps of surfaces

g. Ruled surfaces

h. Elliptic surfaces

i. Classification of surfaces

2. Minimal Model Program

a. Nef, big and ample Q-divisors

b. Cone theorem

c. Contraction theorem

d. Flips and flops

e. Existence and termination of flips in dimension 3

f. Terminal singularities in dimension 3

g. Contraction maps in dimension 3

h. Some effective results in dimension 3

II. Course Goal：We would like introduce some recent developments in algebraic geometry.

III. Course prerequisite： Algebra

IV. Reference material ( textbook(s) )：

a. Kollar, J., Mori, S., Birational Geometry of Algebraic Varieties, Cambridge University Press

b. Beauville , A., Complex Algebraic Surfaces, Cambridge University Press

c. Hacon,C., The minimal model program for log varieties of general type, http://www.math.utah.edu/~hacon/MMP.pdf

V. Grading scheme：

a. Term Paper 70%

b. Presentation and Participation 30%