Small Talk Seminars, Fall 2013
9 Dec 2013. 15.15-16.15, room 3418, Small Talk Seminar, Katharina Heinrich
Title: The Hilbert scheme of twisted cubics
Abstract: One of the few Hilbert schemes that is well-understood is the moduli space of twisted cubics, that is smooth, proper, rational curves of degree 3 in IP^3. The explicit description was done by R. Piene and M. Schlessinger in 1985. I will explain some of their results, do some explicit calculations and, if time permits, explain how the Hilbert scheme is related to the space of Cohen-Macaulay curves that I work on.
2 Dec 2013. 15.15-16.15, room 3418, Small Talk Seminar, Gustav Sædén Ståhl
Title: The Rees algebra of a module
Abstract: The Rees algebra of an ideal is a basic construction that is used for blowing up schemes. There are several ways of generalizing the construction of Rees algebras to more general modules. I will here present a definition along with some related results given in an article by Eisenbud, Huneke and Ulrich. Finally, I will state some new results regarding this Rees algebra and other observations.
25 Nov 2013. 15.15-16.15, room 3418, Small Talk Seminar, Olof Bergvall
Title: Theta characteristics
Abstract: A theta characteristic on an algebraic curve is nothing fancier than a square root of the canonical bundle of the curve. Nevertheless, the theory of theta characteristics has some quite interesesting consequences in different areas of mathematics, ranging from the classical investigation of bitangents of curves to the theory of spin structures.
In this talk I will give an introduction to theta characteristics on algebraic curves via the theory of quadratic forms on a symplectic vector space over the field of two elements. I will also explain how this theory can be applied to some moduli problems.
18 Nov 2013. 15.15-16.15, room 3418, Small Talk Seminar, Daniel Berg,
Title: Toric stacks
Abstract: Toric varieties are simple enough to describe explicitly, yet complicated enough to provide a rich source of interesting examples of algebro-geometric concepts. Toric methods can also be used to study algebraic stacks. Just as toric varieties, toric stacks have explicit descriptions in terms of combinatorics and homogeneous coordinates. This alows us to compute and describe things such as Picard groups, sheaves of differentials, cotangent complexes, blow ups as well as purely stacky constructions, such as root stacks, explicitly.
I will explain some of the concepts above. Depending on your preferences, you may see this talk as an attempt at introducing toric geometry via algebraic stacks or introducing algebraic stacks via toric geometry. Or you may simply see it as an attempt at introducing both, using elementary algebraic geometry.
11 Nov 2013. 15.15-16.15, room 3418, Small Talk Seminar, David Rydh,
Title: The étale topology.
Abstract: I will give a basic introduction to the étale topology. In particular, I'll explain why the étale topology is essential in algebraic geometry. I will also briefly discuss the relation to the Nisnevich topology and explain the building blocks of the Nisnevich topology (étale dévissage).