Algebraic Geometry (NMAG401) - information about the course in winter semester 2017/2018.

Basic information

The contents of the course and other basic information are available in the Student Information System.

The schedule (to be found also in the Student Information System):

  • lectures on Fridays, 12.20pm-1.50pm in room K9,
  • problem sessions on Fridays, 2.00pm-3.30pm in room K9.

Exam

The exam will be oral. Please contact me to agree on the date and time of your exam.

Credit

The credit will be granted for solved exercise problems. There will be three sets of the problems and they will appear here. Requested are at least 50 % of successfully solved problems handed in withing the deadlines.

Problem set #1 (deadline for solutions: November 24)

  1. A complex algebraic subset X ⊆ A2 is called a conic if it is of the form X = V(f) where f ∈ C[x,y] is a non-zero polynomial of total degree 2. Show that every irreducible conic is isomorphic either to V(y-x2) or to V(xy-1).
  2. Let C be the field of complex numbers and consider the ring R = C[x,y,z]/(xz,yz) and the element f := y-z ∈ R. Show that the localized ring Rf (the element f is made invertible) is isomorphic to the ring C[x,y±1] × C[z±1].
  3. Consider the subalgebra R ⊆ C[x,y] which consists of the polynomials only with terms of even total degree. Find an algebraic set X such that R is isomorphic to the coordinate ring C[X].

What has been lectured

A brief overview of what has been taught in individual lectures, including references to the literature, can be found below.

October 6, 2017
Affine algebraic sets, Zariski topology, the decomposition of a noetherian topological space to irreducible components, algebraic subsets of the affine plane (lecture notes, sec. 1; [Ful], sec. 1.2, 1.4 - 1.6, 6.1; [Ga], sec. 1.1, 1.3).
October 13, 2017
The decomposition of a noetherian topological space to irreducible components (proof), the ideal of a subset of an affine space and a characterization of irreducibility via prime ideals, polynomial maps and coordinate rings (lecture notes, sec. 1 and 2; [Ful], sec. 1.3, 1.5, 2.1 - 2.2; [Ga], sec. 1.3, 2.1).
October 20, 2017
The connection between polynomial maps and homomorphisms of coordinate rings, rational functions on varieties and function fields, rational maps between varieties (lecture notes, sec. 2; [Ful], sec. 2.2, 2.4, 6.6; [Ga], sec. 2.1, 2.3; [Sh], ch. 3).
October 27, 2017
The connection between rational maps with dense image and homomorphisms of function fields, birational equivalence, rational varieties (lecture notes, sec. 2; [Ful], sec. 6.6; [Ga], sec. 2.1, 2.3, 4.3; [Sh], ch. 3).
November 3, 2017
Rational plane curves, localization of rings (lecture notes, sec. 2 and 3; [Sh], ch. 3; [AM], ch. 3).
November 10, 2017
A geometric interpretation of the localization K[X] → K[X]f, radical ideals, Hilbert's Nullstellensatz (lecture notes, sec. 3; [Ful], sec. 1.7 - 1.10, 6.3; [Ga], sec. 1.2, 2.1; [AM], ch. 7).

Literature

I will write and update a draft version of lecture notes during the semester: [Full text in PDF]

As far as other sources are concerned, algebraic geometry has gradually grown to a very broad field with thousands of pages written about it from several points of view. To get an impression (or to get an idea where to go next for those interested in the topic), check this blog on mathoverflow.net. The core of the lecture is presented according to the following sources available in PDF:

[Ga] A. Gathmann, Algebraic geometry, notes from a course in Kaiserslautern, 2002/2003. [Full text in PDF]
[Ful] W. Fulton, Algebraic Curves (An Introduction to Algebraic Geometry), 2008. [Full text in PDF]

The lectures may also involve facts from the following (off-line) sources:

[Sh] I. R. Shafarevich, Basic algebraic geometry 1, Varieties in projective space, 2. vyd., Springer-Verlag, Berlin, 1994.
[AM] M. F. Atiyah, I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., 1969.
[CLO] D. Cox, J. Little, D. O'Shea, Ideals, varieties, and algorithms, Second Edition, Springer, New York, 2005.
[Na] M. Nagata, Local Rings, John Wiley & Sons, 1962.
[Nee] A. Neeman, Algebraic and Analytic Geometry, LMS Lecture Note Series 345, Cambridge, 2007.

Other links