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Vorlesung + Übung: Algebraic Geometry II - Details

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Algebraic Geometry II

General information

Course number MTH-1482
Semester SS 2019
Home institute Algebra und Zahlentheorie
Courses type Vorlesung + Übung in category Teaching
First appointment Thu , 25.04.2019 10:00 - 11:30, Room: (L1-3008)
Participants Master Mathematics
Performance record Portfolioprüfung
Hauptunterrichtssprache englisch
Literaturhinweise Cohomology and/or general concepts:
Kempf, Algebraic Varieties
Görtz and Wedhorn, Algebraic Geometry I
Hartshorne, Algebraic Geometry
Mumford, The Red Book of varieties and schemes
Shafarevic, basic algebraic geometry 2

Deformations:
Sernesi, Deformations of algebraic schemes

Surfaces:
Beauville, complex algebraic surfaces
Mumford, lectures on curves on an algebraic surface
Reid, Chapters on algebraic surfaces (available online for free)

Construction of Hilbert, Quot and Picard schemes:
Fantechi et al., Fundamental Algebraic Geometry

Historical perspective:
The Unreal Life of Oscar Zariski, by Carol Parikh
ECTS points 9

Lecturers

Tutor

Times

Thursday: 10:00 - 11:30, weekly (from 25/04/19), Übung
Thursday: 12:15 - 13:45, weekly (from 25/04/19)
Friday: 12:15 - 13:45, weekly (from 26/04/19)

Course location

(L1-3008) Thu.. 10:00 - 11:30 (12x)
(L1-2004) Thu.. 12:15 - 13:45 (12x)
Fri.. 12:15 - 13:45 (14x)

Fields of study

Comment/Description

The course will begin by reviewing (and possibly reinterpreting) the most important definitions and facts about schemes (treated in AG I) and introduce some other central tools (cohomology of sheaves, divisors, invertible sheaves, line bundles, possibly bits of deformation theory).

As a motivation for doing this, we will see how this new language solves some major problems which had remained for decades unsolved in the realm of the classical language of algebraic geometry: the construction of moduli spaces. As sets, these are collections of some specific algebro-geometric objects existing in nature (e.g. the set of curves on a given surface, the set of invertible sheaves on a given scheme, the set of all curves, etc.) up to suitable isomorphism. The question is:

can we give to such a set a "natural" structure of variety/scheme?

(A basic -positive- example is a projective space over a field as the variety parametrising 1-dimensional vector subspaces of a given vector space. A generalisation is given by Grassmannians.)

This question is extremely hard in general (in fact, essentially negative) but can be solved in some specific cases. We are going to focus on one, maybe two of them. With this excuse, we are going to learn about the fascinating theories of curves and surfaces. In some sense, all of the main problems of contemporary algebraic geometry have their roots there.

attendance

Current number of participants 8