Vorlesung + Übung: Algebraic Geometry II - Details

Allgemeine Informationen

Veranstaltungsname	Vorlesung + Übung: Algebraic Geometry II
Veranstaltungsnummer	MTH-1482
Semester	SS 2019
Aktuelle Anzahl der Teilnehmenden	4
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
Veranstaltung findet online statt / hat Remote-Bestandteile	Ja
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

Lehrende

• Marco Ramponi

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)

Studienbereiche

• Mathematisch-Naturwissenschaftlich-Technische Fakultät > Institut für Mathematik > Master Mathematik

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.