Warning: This web page is about the course in winter semester 2013/2014. Please check the home page for information about the current course.
Basic information
The contents of the course and other basic information is available in the Student Information System.
The schedule (to be found also in the Student Information System):
 lectures on Thursdays, 3:405:10pm in room K5,
 problem sessions on Fridays, 12:201:50pm in room K5.
Exam
The exam will be oral and it will take place on:
 Tuesday January 14 at 10am in the seminar room of Dept. Alg. (link to SIS),
 Thursday January 30 at 1pm in the seminar room of Dept. Alg. (link to SIS),
 Wednesday February 12 at 10am in the seminar room of Dept. Alg. (link to SIS).
There will be no other official exam days. If for any reason you do not manage to do the exam in these days, please contact me.
Credit
The credit will be granted for solutions of the tasks below. Requested are at least 50% of successfully solved problems handed in withing the deadlines.
Problem set #1 (deadline for solutions: November 8)
 Let X_{1} and X_{2} be algebraic sets. Show that I(X_{1} ∩ X_{1}) is equal to the redical of I(X_{1}) + I(X_{2}). Find an example where I(X_{1} ∩ X_{1}) ≠ I(X_{1}) + I(X_{2}) .
 Let p be a prime number. Show that the ideal I_{p} = (x^{p1} + x^{p2} + ... + x + 1, x1) in the ring Z[x] is maximal. What is the quotient field Z[x] / I_{p}?
 Let K be an algebraically closed field and I ⊆ K[x_{1}, ..., x_{n}] be an ideal. Show that the radical of I is equal to the intersection of the maximal ideals of K[x_{1}, ..., x_{n}] which contain I.
 Determine generators of the maximal ideal M of the polynomial ring R[x,y] which has zero in (1+i,1i) ∈ A².
Problem set #2 (deadline for solutions: December 13)
 Let φ^{*}: C[s,t]/(t^{4}  s^{3}  s^{2}) → C[x,y]/(y^{2}  x^{3} + x) be a map of Calgebras given by φ^{*}(s) = x^{2}  1 and φ^{*}(t) = y. Show that φ^{*} is a well defined injective homomorphism of Calgebras. Show also that the corresponding polynomial map φ: X → Y is almost a bijection in the sense that, up to one exception, every point of Y has a unique preimage.
 A Calgebraic subset X ⊆ A^{2} is called a conic if it is of the form X = V({f(x,y)}) where f(x,y) is a nonzero polynomial of total degree 2. Show that every irreducible conic is isomorphic either to V({yx^{2}}) or to V({xy1}).
 Consider the Calgebraic subset X = V({y^{2}  x^{3}}) of A^{2} and the regular function r: X \ {(0,0)} → A^{1} given by r(a,b) = a/b. Is it possible to extend r to a regular function defined everywhere on X? Why?
Problem set #3 (deadline for solutions: January 9)

Let K be an algebraically closed field. We shall define a map i from the set of 2dimesional subspaces V ⊆ K^{4} to the projective space P^{5} as follows. We arbitrarily choose two linearly independent vectors v = (a_{0}, a_{1}, a_{2}, a_{3}) and w = (b_{0}, b_{1}, b_{2}, b_{3}) in V, we write the components into a 2x4 matrix
a_{0} a_{1} a_{2} a_{3} b_{0} b_{1} b_{2} b_{3} and we compute (in any fix order) the 6 determinants d_{0}, d_{1}, ..., d_{5} of all 2x2 submatrices. Then we define i to map V to (d_{0} : d_{1} : ... : d_{5}).
 Show that i is a well defined injective map.
 Show that the image G of i is a projective variety given in P^{5} by a single quadratic equation.
 Show that we can cover G (as an abstract variety) by 6 open subsets isomorphic to A^{4}.
(Hint: Apply Gauss elimination to the 2x4 matrix.)
 Consider the 6dimensional subspace V of C[x,y] consisting of all polynomials of total degree at most 2. Consider also the corresponding projective space P^{5} whose elements are lines in V. Show that the irreducible conics in the complex affine space A^{2} are parametrized by a Zariski open subset of P^{5}.
What has been lectured
Below is a brief overview of what has been taught in individual lectures.
 October 3, 2013
 Algebraic sets, the assignments V and I, Zariski topology, varieties and the decomposition to irreducible components.
 October 10, 2013
 The bijection between polynomial maps of algebraic sets and homomorphisms of coordinate rings.
 October 17, 2013
 Isomorphisms of algebraic sets, localizations of rings, preparation for the Nullstellensatz.
 October 24, 2013
 Hilbert's Nullstellensatz, the spectrum and maximal spectrum of a commutative ring, for algebraically closed fields the identification of an algebraic set with the maximal spectrum of its coordinate ring.
 October 31, 2013
 For algebraically closed fields the translation of the Zariski topology and polynomial maps to the max. spectrum, for general fields a version of this correspondence using the orbits of the Galois groups of the algebraic closure of K over K.
 November 8, 2013
 Regular functions and the description of them on the standard basis for the Zariski topology.
 November 14, 2013
 Geometric interpretation of localization K[X]_{f} where K[X] as a coordinate ring, and the proof that regular functions are continuous. The ringed space structure of an affine algebraic set.
 November 21, 2013
 Sheaves of algebras, ringed spaces and maps between them. Basic open subspaces of affine algebraic sets are affine.
 November 28, 2013
 Finishing the proof about basic open subsets of algebraic sets. Abstract prealgebraic sets and prevarieties. Gluing two ringed spaces along a common open ringed subspace.
 December 5, 2013
 A general version of gluing of ringed spaces, products of abstract prealgebraic sets, abstract algebraic sets and the criterion with the diagonal of Δ(X) of X×X.
 December 12, 2013
 Projective algebraic sets, noetherian topological spaces, a general version of decomposition to irreducible components.
 December 19, 2013
 Projective Nullstellensatz, projective algebraic sets as ringed spaces and abstract algebraic sets.
 January 3, 2014
 Closedness of morphisms from projective algebraic sets, Krull dimension (first steps).
 January 9, 2014
 Krull dimension of affine and projective spaces computed, introduction to local rings at points.
Literature
This is a recently introduced course and with no lecture notes available specifically for it. The core of the lecture is nevertheless 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 also involve some facts from commutative algebra which can be found in the following (offline) sources:
[AM]  M. F. Atiyah, I. G. MacDonald, Introduction to Commutative Algebra, AddisonWesley 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. 