Advanced Algebraic Geometry - R - 8EC

Course:
Advanced Algebraic Geometry: Moduli spaces of sheaves


Prerequisites:
Basics of varieties, schemes, coherent sheaves and sheaf cohomology, as in the Mastermath course Algebraic Geometry II, or e.g. Chapters I-III of the book Algebraic Geometry by Hartshorne, plus Riemann-Roch on curves.


Aim of the course:
A moduli space of sheaves is a space whose points parametrize sheaves on a fixed variety. The moduli space is a powerful tool to study these sheaves, and moreover it has very interesting geometric properties. The goal of this course is to show that under the right conditions, moduli spaces of sheaves exist, and to study their geometry, both abstractly and in examples.

After reviewing some basic properties of vector bundles, coherent sheaves and homological algebra, we will discuss various examples of stable bundles, which are interesting in their own right. We then proceed to discuss the construction of the moduli space and what can be said about universal families on them. The construction builds upon Grothendieck's Quot schemes, which are interesting and indispensable in the study of moduli problems in general. Finally, we will focus on examples, e.g. the case of curves and K3 surfaces.

Homework & Exams

There will be exercises every week. Roughly every other week, some of them should be handed in as homework. The homework counts for 30 percent of the final grade. The remaining 70 percent is determined by the final exam, which will be oral or written depending on the number of students. To pass the course, students need to score a minimum of 5.0 in the final exam and a 5.5 for the weighted average of homework and exam. The resit exam will give an opportunity to improve the exam score only; there is no second chance for the homework. Even when taking the resit exam the homework will count towards the final grade.

Literature:

The main reference is the book The geometry of moduli spaces of sheaves by D. Huybrechts and M. Lehn, Cambridge University Press, 2nd edition, 2010. More references will be announced during the course.


Lecturers:
Emma Brakkee (Leiden) and Yajnaseni Dutta (Leiden)