Topics in Arithmetic Geometry, I

代數幾何專題 I

Course Information

Instructor : Jungkai Chen陳榮凱, I-Hsun Tsai蔡宜洵

Place: Old Math Bldg 103

Hours: Tue. 9:10-12:10

 

First meeting

Sep. 20 (Tue.) 9:10

 

Syllabus

The two main theme of this course are minimal model theory and etale cohomology.

There are some important recent developments in the theory of minimal model using the technique of multiplier ideals and Shokurov’s work. In order to introduce these developments, we would like to review previous work on minimal model theory in this course.

Another main theme is etale cohomology. Etale cohomology can be considered a cohomology theory that bridge the arithmetic side and geometric side. It’s now considered as a fundamental tool for arithmetic geometry. For example, one can prove Weil conjecture which concerns number of points on varieties.

 

Topics for coming weeks:

        Sep. 20    Overview of Minimal Model Program and Etale Cohomology

        Sep. 27    Existence of rational curve when KX is not NEF

                        Etale morphism

        Oct. 4      Cone Theorem

                        Etale topology

        Oct. 11     Non-vanishing theorem and base point freeness theorem

                        Sheaves for the etale topology

        Oct. 18   Contraction theorem

                        Etale cohomology

 

Prerequisite

Any ambitious student is welcome.

The potential speakers are expected to have general background in algebra, topology, complex and algebraic geometry. The latter should include the language of schemes and their morphisms as in Hartshorne's book.

 

Reference:

Freitag, Kiehl, Etale Cohomology and the Weil Conjecture, Springer 1988

Milne, Etale Cohomology, Princeton 1980

Milne, Lectures on Etale Cohomology, http://www.jmilne.org/math/CourseNotes/math732.pdf

Tamme, Introduction to Etale Cohomology, Springer, 1994.

Kiehl, Weissauer, Weil Conjectures, Perverse Sheaves and l’adic Fourier Transform, Springer, 2001

Clemens, Kollar, Mori, Higher dimensional complex geometry, Asterisque 166, SMF 1988.

Kollar, Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math 134. 1998.

Matsuki, Introduction to Mori Program, Springer, 2001.

 

Grading:

        Presentation      30 %

        Homework       30 %

        Term Paper      40 %