Masters Seminar in Algebraic Geometry

This course is a seminar for Masters or advanced Bachelor students; one can receive credit as either a Bachelor or Masters seminar. The main topic of the seminar is the enumerative geometry, a subject within algebraic geometry. The goal of enumerative geometry is to count how many solutions there are to a well-defined geometric problem. As elementary example: one takes a circle C in the plane and a point p outside of C. How many lines through p are tangent to C? The answer, which one can see without making any calculations, is obviously 2. A more difficult question is: let L1,..., L5 be lines in general position in the plane. How many conics are tangent to all five lines? To answer these and similar questions, one applies various basic principles of algebraic geometry, which are also in themselves very interesting. The main goal of this seminar is to learn and apply these basic principles.

As prerequisites, one should have taken at least Algebra 1 and 2. The first semester of algebraic Geometry, or its equivalent, would be helpful but not absolutely necessary.

Preliminary list of topics

1. The affine and projective planes, curves in the plane, intersection points and intersection numbers, Bezout's theorem.
2. Affine space and projective space, hypersurfaces, intersection points and intersection numbers.
3. linear subspaces of projective space: the Grassmann variety. Plücker coordinates and relations, Schubert cells, Schubert varieties and the Schubert calulus.
4. Proper and improper intersections: regularizing by moving and by blowing up.

Instructor: Marc Levine

Schedule: Fridays, 14-16 Uhr, WSC-S-U-3.02. The first meeting is on 21.10.2016.

Interested students who have a schedule conflict should send an email to Marc Levine.

Outline Download.


Lecture 1. (Marc Levine) Affine algebraic sets and affine algebraic varieties.Chapters 1 and 2 [AC].

Lecture 2. (Yukihide Nakada) Local properties of plane curves: multiple points, tangent lines, multiplicities and local rings,
Chap. 3, §1,2, Problems 3-5, 3-9, 3-11, 3-13 [AC].

Lecture 3. (Jonas Franzel) Local properties of plane curves: intersection multiplicities.
Chap. 3, §3, problems 3-17, 3-19 [AC].

Lecture 4. (Chirantan Chowdhury) Projective space and projective varieties.
Chap. 4, §1-3. Problems 4-2, 4-12, 4-15 [AC].

Lecture 5. (Tim Inoue) Projective plane curves: definitions and linear systems.
Chap. 5, §1,2, Problems 5-1, 5-7, 5-17 [AC].

Lecture 6. (Jonas Franzel) Projective plane curves: Bezout's theorem, multiple points.

Chap. 5, §3,4. Problems 5-25, 5-26 [AC].

Lecture 7. Projective plance curves: Max Noether's fundamental theorem, applications.
Chap. 5, §5,6. Problems: choose two or three from 5-32 through 5-38 [AC].

Lecture 8. Applications of Bezout's theorem:
a. Topological genus of a smooth curve in CP2. Discuss: projection from a point in P2, the Riemann surface of a smooth curve in CP2, local description of maps of Riemann surfaces, triangulations and the genus formula.
b. Counting tangent lines: the dual curve of a projective curve in P2, counting flexes and multiple tangents, counting common tangent lines to two curves.

Program Download.


[AC] W. Fulton, Algebraic Curves. An introduction to algebraic geometry. Notes written with the collaboration of Richard Weiss. Mathematics Lecture Notes Series. W. A. Benjamin, Inc., New York-Amsterdam, 1969. xiii+226 pp.


Algebraic Curves, Chap. 1, 2

Algebraic Curves, Chap. 3

Algebraic Curves, Chap. 4, 5