Because of the attack on the computer network of the University of Duisburg-Essen, the ESAGA site currently can be reached only at https://www.esaga.net/.... Links pointing to https://www.esaga-uni-due.de/ do not work (but one could manually replace uni-due.de by net), and similarly for content, such as images, at such addresses contained in our pages. (Email messages to @uni-due.de addresses are accepted by our mail servers again.)

We apologize for the inconvenience caused by these problems.

# RTG-Seminar

The Thursday morning seminar (10:15-11:45 in WSC-N-U-3.05) will be the “Research Training Group Seminar” where members of the RTG (PhD students, post-docs,…) present their results. Sometimes, we also have speakers from other places. Depending on the number of speakers and on the proposed topic, a speaker could use one or two sessions.

Termin Vortragende*r Titel
20.10.2022 RTG General Assembly
27.10.2022 Dario Weißmann A functorial approach to the stability of vector bundles
3.11.2022 V. Srinivas (Tata Institute) Grothendieck-Lefschetz and Noether-Lefschetz for the divisor class group
10.11.2022 Georg Linden (Wuppertal) Equivariant vector bundles on the Drinfeld upper half space
17.11.2022 V. Srinivas (Tata Institute) Grothendieck-Lefschetz and Noether-Lefschetz for the divisor class group
24.11.2022 V. Srinivas (Tata Institute) Grothendieck-Lefschetz and Noether-Lefschetz for the divisor class group
Mon, 28.11.2022, 4-6pm
S-U-3.03
V. Srinivas (Tata Institute) Grothendieck-Lefschetz and Noether-Lefschetz for the divisor class group
1.12.2022 Dan Clark (Durham/ Münster) The Geometry of the Unipotent component of the moduli space of Weil-Deligne Representations
12.1.2023 Bence Forrás Integrality of smoothed $p$-adic Artin $L$-functions
19.1.2023 reserviert
26.1.2023 reserviert
2.2.2023 N. N. tba

## Abstracts

### Dario Weißmann: A functorial approach to the stability of vector bundles

Semistability is a property of vector bundles which is functorial under pullback by a finite separable morphism, but this is no longer the case for stability. However, the general stable bundle on a smooth projective curve remains stable after pullback by all finite separable morphisms which are prime to the characteristic. Furthermore, this property defines a large open in the moduli space of bundles. In contrast, a result of Ducrohet and Mehta states that the etale trivializable bundles are dense in the moduli space of bundles, i.e., avoiding the characterstic is essential.

### V. Srinivas (Tata Institute): Grothendieck-Lefschetz and Noether-Lefschetz for the divisor class group

Let $X$ be a normal projective variety over an algebraically closed field of characteristic $0$, and $f\colon X\to {\mathbb P}^n$ a finite morphism. Let $Y=f^{-1}(H)$ be the inverse image of a general hyperplane section. In these lectures, I will sketch proofs (obtained with G. V. Ravindra) of results on the map of divisor class groups ${\rm Cl}(X)\to {\rm Cl}(Y)$, analogous to the “classical” Grothendieck-Lefschetz and Noether-Lefschetz theorems for the Picard groups of smooth projective varieties. As an application, we discuss Kollar’s results on recognizing normal projective varieties from the underlying Zariski topological spaces.

### Dan Clark (Durham/ Münster): The Geometry of the Unipotent component of the moduli space of Weil-Deligne Representations

Let $G$ be a split connected reductive group with Lie group $\mathfrak{g}$ and set $q>1$ an integer. Define the Scheme / O $$S_G( R)= \{(\Phi,N)\in G( R)\times\mathfrak{g}( R)|Ad(\Phi).N=qN\}.$$ This can be interpreted as the unipotent connected component of the moduli space of Weil-Deligne representations valued in G. In this talk, we explore some of the local properties of the irreducible components of this scheme, and of a certain union of irreducible components, and prove that at most points, the subscheme is Cohen-Macaulay.