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Information on this page is given in the language of instruction: English, German, or French.
Description: In this lecture, we continue to build a toolbox for the study of algebraic varieties and, more generally, schemes and sheaves. We start by reviewing important constructions of schemes, such as the glueing of schemes, the fiber product of two schemes over a base, and the relative projectivization of a sheaf over a scheme. We use these constructions to introduce two procedures, that can be thought of as ways to ``desingularize’’ a singular scheme: normalization and blowing up. As these procedures are characterized by universal properties, we will spend some time describing and analysing various types of morphisms between schemes, many motivated by our topological intuition: open immersions, closed immersions, proper morphisms, projective morphisms, etc… This part of the lecture will conclude on a light introduction to positivity notions — such as ampleness, nefness, and effectivity — for Cartier divisors and locally free sheaves of rank 1 in algebraic geometry. We then focus on constructing and understanding the cohomology of sheaves on noetherian schemes. Among the highlights of this part are Serre vanishing, Serre duality, the adjunction formula and, if time allows, the semicontinuity theorem.
Time allotment: 2 SWS + 2 SWS Übungen by Yi-Hung Chen. This lecture can be validated jointly with the Seminar Topics in Algebraic Geometry as one class (4 SWS + 2 SWS Übungen).
Language of instruction: English.
Prerequisites: Algebraische Geometrie I.
References: Hartshorne, Algebraic Geometry (especially Chapters II and III).
Practicalities: Moodle-Link. Fridays 8-10 in Room IA1/177.
Description: This seminar, taught jointly with Prof. Ivanov, can either be validated on its own or jointly with Algebraische Geometrie II as one class. The seminar will provide examples and applications of the tools usually developed in algebraic geometry lectures. The focus is on the geometry of curves and surfaces over various fields.
Practicalities: Moodle-Link. If you wish to attend and give a talk, please email me and Prof. Ivanov timely. Fridays 10-12 in Room IA1/181.