Algebraic Curves | Dali Shen's Homepage

Algebraic Curves (Fall 2024)

What is this course about?

Algebraic geometry is the study of solutions of polynomial equations, and has close connections with other areas of mathematics.

In this course, I will give a basic introduction to algebraic varieties, focusing on using curves (the dimension one case) to illustrate the basic notions in algebraic geometry.

Prerequisite

Basic knowledge of polynomials, rings and ideals.

Literature

[1] E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of algebraic curves, Vol. I, Grundlehren Math. Wiss., 267, Springer-Verlag, New York, 1985. xvi+386 pp.

[2] W. Fulton, Algebraic curves, An introduction to algebraic geometry, Notes written with the collaboration of Richard Weiss, Reprint of 1969 original, Adv. Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. xxii+226 pp.

[3] Phillip A. Griffiths, Introduction to algebraic curves, Translated from the Chinese by Kuniko Weltin, Transl. Math. Monogr., 76, American Mathematical Society, Providence, RI, 1989. x+221 pp.

Place and time

Mondays, 13:30-15:05; Venue: A3-1a-204

Wednesdays, 09:50-11:25; Venue: A3-1a-204

ZOOM: 559 700 6085, PW: BIMSA

Starting from Sep. 23

WeChat Group

We have created a WeChat Group for communication. You could write to me if you would like to join.

Material covered

Sep. 23: Introduction with some examples.

Sep. 25: Some more examples.

Sep. 30: Algebraic preliminaries.

Oct. 2: Holiday, no class.

Oct. 7: Holiday, no class.

Oct. 9: Algebraic sets, the ideal of a set, Hilbert Basis Theorem.

Oct. 14: Irreducible components, algebraic subsets of the plane.

Oct. 16: Hilbert's Nullstellensatz.

Oct. 21: Coordinate rings, polynomial maps, coordinate changes.

Oct. 23: Rational functions and local rings, discrete valuation rings, forms.

Oct. 28: Ideals with a finite number of zeros, exact sequence.

Oct. 30: Multiple points and tangent lines.

Nov. 4: Multiplicities and local rings.

Nov. 6: Intersection numbers.

Nov. 11: Projective space.

Nov. 13: Projective algebraic sets.

Nov. 18: Affine and projective varieties, notions for projective plane curves.

Nov. 20: Linear systems of curves, Bézout's Theorem.

Nov. 25: Multiple points, Max Noether's Fundmental Theorem.

Nov. 27: Applications of Noether's Theorem.

Dec. 2: The Zariski topology, varieties.

Dec. 4: Morphisms of varieties, products and graphs.

Dec. 9: Dimension of varieties, rational maps.

Dec. 11: Blowing up.

Dec. 16: Riemann's theorem.

Dec. 18: Riemann-Roch theorem.

There is also a webpage at BIMSA dedicated to this course where you could find all the videos for this course.