SAG
Seminar Algebraic Geometry Autumn 2023
This is a weekly research seminar in algebraic geometry organized by
Laura Pertusi,
Mirko Mauri,
Raymond Cheng, and
Sarah Frei as part of the
Junior Trimester Program
in Algebraic Geometry at the Hausdorff Research Institute for
Mathematics. Meetings take place from 10:30 to 11:30, usually on Thursdays, in
the Hausdorff Institute Lecture Hall, Poppelsdorfer Allee 45.
- 28.09
- Noah Olander
- Fully faithful functors and dimension
- Can one embed the derived category of a higher dimensional variety into the derived category of a lower dimensional variety? The expected answer was no. We give a simple proof and prove new cases of a conjecture of Orlov along the way.
- 12.10
- Genki Ouchi
- Cubic fourfolds and K3 surfaces with large automorphism groups
- Relations between cubic fourfolds and K3 surfaces are described by Hodge theory and derived categories. Using Hodge theory and derived categories, we can show that cubic fourfolds and associated K3 surfaces share their symmetries, which are related with Mathieu groups and Conway groups. In this talk, we find pairs of a cubic fourfold and a K3 surface sharing large symplectic automorphism groups via Bridgeland stability conditions on K3 surfaces.
- 02.11
- Franco Rota
- Non-commutative deformations and contractibility of rational curves
- When can we contract a rational curve \(C\)? The situation is much more complicated for threefolds than it is for surfaces: Jimenez gives examples of (-3,1)-rational curves neither contract nor move. Their behaviour is controlled by the functor of non-commutative deformations of \(C\), which conjecturally controls exactly their contractibility. I will report on work in progress with M. Wemyss, and reinterpret some of Jimenez's examples in terms of non-commutative deformations.
- 16.11
- Andres Fernandez Herrero
- Towards curve counting on the classifying stack \(B\mathrm{GL}_n\)
- In this talk I will describe a version of stable maps into a quotient stack \([Z/\mathrm{GL}_N]\), where \(Z\) is a projective variety with an action of the general linear group \(\mathrm{GL}_N\). If time allows, I will also update on the ongoing piece of the story with marked points, which involves some surprises such as the inclusion of a notion of `orientation' for the markings in order to compactify the evaluation morphisms and recover reasonable gluing morphisms. This talk is based on joint work in progress with Daniel Halpern-Leistner.
- 27.11
- Olivier Martin
- Isotrivial Lagrangian fibrations of compact hyper-Kähler manifolds
- I will present a recent exploration of the geometry of isotrivial Lagrangian fibrations conducted with Y. Kim and R. Laza. We show that the smooth fiber of such a fibration is isogenous to the power of an elliptic curve. Moreover, we introduce a dichotomy arising from the Kodaira dimension of the minimal Galois cover of the base trivializing monodromy. We also classify up to birational equivalence isotrivial fibrations with a section in the case where this cover has Kodaira dimension 0.
- 07.12
- Roberto Fringuelli
- The automorphism group of the moduli space of \(G\)-bundles over a curve
- For any almost-simple group \(G\) over an algebraically closed field \(k\) characteristic zero, we describe the automorphism group of the moduli space of semistable \(G\)-bundles over a curve of genus at least 4. The result is achieved by studying the singular fibers of the Hitchin fibration. Time permitting, we also provide a proof of a Torelli-type theorem for these moduli spaces.