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Vorlesung + Übung: Algebraic Geometry II - Details
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Lehrveranstaltung wird online/digital abgehalten.

Allgemeine Informationen

Veranstaltungsname Vorlesung + Übung: Algebraic Geometry II
Veranstaltungsnummer MTH-1482
Semester SS 2019
Aktuelle Anzahl der Teilnehmenden 6
Heimat-Einrichtung Algebra und Zahlentheorie
Veranstaltungstyp Vorlesung + Übung in der Kategorie Lehre
Erster Termin Donnerstag, 25.04.2019 10:00 - 11:30, Ort: (L1-3008)
Teilnehmende Master Mathematics
Leistungsnachweis Portfolioprüfung
Online/Digitale Veranstaltung Veranstaltung wird online/digital abgehalten.
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-Punkte 9

Räume und Zeiten

(L1-3008)
Donnerstag: 10:00 - 11:30, wöchentlich (12x)
(L1-2004)
Donnerstag: 12:15 - 13:45, wöchentlich (12x)
Freitag: 12:15 - 13:45, wöchentlich (14x)

Kommentar/Beschreibung

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.