Algebraic Geometry (NMAG401) - information about the course in winter semester 2013/2014.

## 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, 3:40-5:10pm in room K5,
• problem sessions on Fridays, 12:20-1:50pm 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)

1. Let X1 and X2 be algebraic sets. Show that I(X1 ∩ X1) is equal to the redical of I(X1) + I(X2). Find an example where I(X1 ∩ X1) ≠ I(X1) + I(X2) .
2. Let p be a prime number. Show that the ideal Ip = (xp-1 + xp-2 + ... + x + 1, x-1) in the ring Z[x] is maximal. What is the quotient field Z[x] / Ip?
3. Let K be an algebraically closed field and I ⊆ K[x1, ..., xn] be an ideal. Show that the radical of I is equal to the intersection of the maximal ideals of K[x1, ..., xn] which contain I.
4. Determine generators of the maximal ideal M of the polynomial ring R[x,y] which has zero in (1+i,1-i) ∈ A².

### Problem set #2 (deadline for solutions: December 13)

1. Let φ*: C[s,t]/(t4 - s3 - s2) → C[x,y]/(y2 - x3 + x) be a map of C-algebras given by φ*(s) = x2 - 1 and φ*(t) = y. Show that φ* is a well defined injective homomorphism of C-algebras. 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.
2. A C-algebraic subset X ⊆ A2 is called a conic if it is of the form X = V({f(x,y)}) where f(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}).
3. Consider the C-algebraic subset X = V({y2 - x3}) of A2 and the regular function r: X \ {(0,0)} → A1 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)

1. Let K be an algebraically closed field. We shall define a map i from the set of 2-dimesional subspaces V ⊆ K4 to the projective space P5 as follows. We arbitrarily choose two linearly independent vectors v = (a0, a1, a2, a3) and w = (b0, b1, b2, b3) in V, we write the components into a 2x4 matrix

 a0 a1 a2 a3 b0 b1 b2 b3

and we compute (in any fix order) the 6 determinants d0, d1, ..., d5 of all 2x2 submatrices. Then we define i to map V to (d0 : d1 : ... : d5).

1. Show that i is a well defined injective map.
2. Show that the image G of i is a projective variety given in P5 by a single quadratic equation.
3. Show that we can cover G (as an abstract variety) by 6 open subsets isomorphic to A4.

(Hint: Apply Gauss elimination to the 2x4 matrix.)

2. Consider the 6-dimensional subspace V of C[x,y] consisting of all polynomials of total degree at most 2. Consider also the corresponding projective space P5 whose elements are lines in V. Show that the irreducible conics in the complex affine space A2 are parametrized by a Zariski open subset of P5.

## 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 (off-line) sources:

 [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.