Hugo Zock

Address: Universitätstraße 1, 40225 Düsseldorf

Office: 25.22.03.51

Since September 2024, I am a PhD student in the working group of Stefan Schröer at the university of Düsseldorf. My second advisor is Kay Rülling at the university of Wuppertal. I am funded by the Research Training Group GRK2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology. From 2022 to 2024 I was a student of the ALGANT master program, spending my first year at the university of Leiden, and my second year at the university of Regensburg. In Regensburg I wrote a thesis titled "A local Version of Kashiwara's Conjecture" under the supervision of Moritz Kerz (linked below).

Broadly speaking, I am interested in algebraic geometry in positive and mixed characteristic.

My CV.

Some things I have written

Below you will find some unpublished writing of mine that I think might be interesting for some people that come across it.

• "A Variety with trivial canonical Bundle, degenerate Hodge-to-de Rham spectral Sequence, and obstructed Deformations"

  Abstract: We exhibit a variety as in the title, resolving a question of Brantner and Taelman.

• "Bloch-Kato ordinary bielliptic Surfaces"

  Abstract: We determine the Bloch-Kato ordinary (quasi-)bielliptic surfaces in every characteristic. We relate the notion of Bloch-Kato ordinarity to the notions of ordinary, classical and supersingular introduced for (quasi-)bielliptic surfaces by Ivo Kroon.

• "Étale fundamental Groups of nodal Curves in positive Characteristic"

  Abstract: We relate the étale fundamental group of a nodal curve to the étale fundamental group of its normalization. Combined with known results on étale fundamental groups of smooth curves in positive characteristic, this gives us a good grip on the fundamental groups of these nodal curves.

• "A Spreading Argument for l-adic Representations with finite Determinant"

  Abstract: We improve a classical spreading argument for l-adic representations allowing us to preserve the property of having finite determinant. Additionally, we prove that such spreadings are unique in an appropriate sense.

• "Twists"

  Abstract: We define the notion of a “base extension”, an abstract framework to axiomatize the notion of Galois descent in various contexts. We subsequently retrieve the well known principle that twisted forms of a k-object X are parametrized by the Galois cohomology set H1(k, Aut(Xs)) for practically all types of objects.

Talks

• "Bloch-Kato ordinary bielliptic surfaces", in Kay Rülling's crystalline cohomology course at the university of Wuppertal, January 2025. My notes for the lecture.
• "Reductive groups over finite fields", GRK2240 Workshop on Deligne-Lusztig varieties, December 2024. My notes for the talk.
• "The Fano Scheme of Lines", Oberseminar Algebraische Geometrie, University of Düsseldorf, October 2024. My notes for the talk.
• "A local Version of Kashiwara's Conjecture", AG-Seminar of prof. M. Kerz, University of Regensburg, June 2024.
• "Brauer-variëteiten met een rationaal punt", Leids Bachelorseminarium, University of Leiden, May 2022.
• "Groepencohomologie en twists van algebra's", Leids Bachelorseminarium, University of Leiden, March 2022.

Where I have been

• Algebraic Geometry in Hannover, March 2025.
• Modern Methods in Moduli, Luxembourg, November 2024.
• Belgian Dutch Algebraic Geometry Seminar, Antwerpen, October 2024.
• (Unofficially) I was at the School on Hodge Theory and Shimura Varieties in Essen in September 2024.

Where I will be

Seminars

• In April of 2024 I hosted the Seminar Mathematical Communication (oder Mathematische Kommunikation) jointly with Lukas Prader at the university of Regensburg. Below are some topics I proposed to students at the time. Many of them are suitable as a starting point for a bachelor's thesis. Feel free to contact me if you are a student at the HHU interested in one of these topics!
  • Quaternions and Conics
  • Every Group is a Fundamental Group
  • Fundamental Groups of Riemann Surfaces
  • Minkowski's Theorem
  • 27 Lines on a Cubic Surface
  • Grothendieck's Galois Theory
  • Kummer Theory
• Tussen 2021 en 2023 was ik studentassistent voor het Leids Seminarium Presenteren & Communiceren. Te zijner tijd heb ik het onderwerp Quaternion en Kegelsneden voorgelegd.

Teaching

• Lineaire Algebra I, Universiteit Leiden, najaar 2020.
• Algebra I, Universiteit Leiden, voorjaar 2021.
• SPC, Universiteit Leiden, najaar 2021-voorjaar 2023.
• Mathematical Communication, University of Regensburg, April 2024.

Miscellaneous

• My master's thesis written under the supervision of Moritz Kerz at the university of Regensburg.
• Notes for a talk I gave on flat families for the seminar course Cohomology of Sheaves and Schemes at the university of Regensburg.
• Notes for a talk I gave on topological groups for the seminar course Fourier Analysis and Representation Theory at the university of Regensburg.
• At some point I wrote an exercise about the Brauer-Hasse-Noether Theorem for function fields for a course involving Galois cohomology.
• Notes for a talk I gave on smoothness for the seminar course Topics in algebraic Geometry at the university of Leiden.
• Notes for a talk I gave on flatness for the seminar course Topics in algebraic Geometry at the university of Leiden.
• Als student in Leiden heb ik ooit een aardig artikeltje over de stelling van Minkowski geschreven voor het vak SPC.