We describe what algebraic geometry is:
- We study algebraic manifolds (point sets fulfilling equations, like function graphs or zero-point sets). In most cases we are interested in zero-sets of multidimensional polynomials
- Our studies depend on the chosen field, Körper (Algebra).
Point sets can somehow define affine spaces (?).
We define projective spaces by setting points that differ by multiplications to be equal.
We classify polynomial in different kinds (e.g. do they contain summands which are products of different dimensions?). They might have different properties later on.
Wir definieren eine Zariski-Topologie (eine sehr grobe Topologie).
There is a theorem that tells us that affine algebraic manifolds over a field correspond with ideals over the same field. (Nehme das Polynom mit den Nullstellen?)
Scheme
What is a scheme? A technique? It seems to be a sort of parallel theory which shares many characteristics with algebraic geometry and some consider it to be the same theory.
We present some of the power of schemes: It seems like they take in a field and spit out something else denoted by .
Blow up
We define a blow-up. If I understand correctly this is an operation to remove crossing points of a manifold be embedding them into higher dimensional space. (There is a person called Hironaka who won a fields medal on this subject)
McKay correspondence
We take a finite subgroup of and do some stuff. We do some incidence stuff and draw a graph describing crossings of spaces. We do some more graph stuff. (Actually this seems quite interesting)
Yasuda (from Osaka) gave a generalization of the McKay correspondence. We next describe some open problems on that correspondence. We mention something about クレパント deletions. (?) We also ask the question whether we can generalize the theorem of Batyrev.