# Northern German Algebraic Geometry Seminar

## 9th - 10th February 2023

### Speakers

 Enrico Arbarello (Rome) Gerard van der Geer (Amsterdam) Daniel Huybrechts (Bonn) Joshua Lam (HU Berlin) Rahul Pandharipande (ETH Zürich) Stefan Schreieder (Hannover) Domenico Valloni (Hannover) Claire Voisin (Paris)

### NoGAGS

The Northern German Algebraic Geometry Seminar is a regular joint seminar of the algebraic geometry groups in Berlin, Bielefeld, Hamburg, Hannover, Leipzig and Oldenburg. Information on the previous meetings can be found here.

### Prize Winners

Two prizes will be awarded on this ocassion. They are:
• the Algebraic Geometry Prize, for the paper
Enrico Arbarello, Andrea Bruno, Edoardo Sernesi: On hyperplane sections of K3 surfaces. Algebraic Geometry 4 (2017), pp 562-596;
• the Compositio Prize (ex aequo), for the paper
Daniel Huybrechts: The K3 category of a cubic fourfold. Compositio Mathematica 153 (2017), pp 586-620.

### Schedule

 Thursday, 9th February 2023 Room 0'119 (Konferenzraum), Erwin Schrödinger-Zentrum 11:00 - 12:00 Rahul Pandharipande Cycles on the moduli space of abelian varieties (notes) Room 0'110, Erwin Schrödinger-Zentrum 13:00 - 14:00 Domenico Valloni Reduction mod $$p$$ of the Noether problem 14:30 - 15:30 Gerard van der Geer Modular forms, moduli of curves and invariant theory 16:00 - 17:00 Stefan Schreieder A moving lemma for cohomology with support 17:15 - 18:15 Bernd SturmfelsOpportunities for young algebraic geometers Friday, 10th February 2023 Room 1.013, Haus 1, Johann von Neumann-Haus 10:00 - 11:00 Claire Voisin Cycle classes on abelian varieties and the geometry of the Abel-Jacobi map 11:30 - 12:30 Joshua Lam Motivic local systems on curves Room 0'115, Erwin Schrödinger-Zentrum 14:00 - 14:30 Award of the AG and Compositio Prizes 14:30 - 15:30 Enrico Arbarello Brill-Noether-Petri curves and K3 curves (notes) 16:00 - 17:00 Daniel Huybrechts The K3 category of a cubic fourfold — an update

### Abstracts

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Enrico Arbarello Brill-Noether-Petri curves and K3 curves

The talk describes results, by various authors, connected to the circle of ideas studied jointly with Edoardo Sernesi and Andrea Bruno in two articles, which appeared in “Algebraic Geometry" in 2014 and 2017.

Gerard van der Geer Modular forms, moduli of curves and invariant theory

We use effective divisors on projectivized Hodge bundles over moduli spaces of curves to construct modular forms. We show that invariant theory enables us to construct all vector valued Siegel modular forms of degree two and three from from such basic modular forms. This is joint work with Fabien Cléry, Carel Faber and Alexis Kouvidakis.

Daniel Huybrechts The K3 category of a cubic fourfold — an update

In this talk I will provide an update about what is known and what is not about the K3 category naturally associated with a smooth cubic fourfold.

Joshua Lam Motivic local systems on curves

By a local system on a complex curve, we mean simply a representation of its fundamental group. It is easy to write down examples of such, and in most cases they come in positive dimensional families. There are certain very special local systems, known as "motivic" or of "geometric origin", which have several favourable properties: these are the local systems which arise in the cohomologies of families of algebraic varieties. For example, such local systems have coefficients in a number field, and are isolated in moduli space. I will discuss some new results in this subject, such as examples where one can prove that motivic local systems are scarce using Hodge theory, as well as an example where one can construct infinitely many rank two local systems on a fixed curve. Some of this is joint work in progress with Daniel Litt.

Rahul Pandharipande Cycles on the moduli space of abelian varieties

I will present a new set of results and conjectures from the perspective of Noether-Lefschetz theory about cycles on the moduli space $$A_g$$ of PPAVs. Joint work with S. Canning and D. Oprea.

Stefan Schreieder A moving lemma for cohomology with support

We prove a moving lemma for cohomology classes with support on smooth quasi-projective varieties with a smooth projective compactification. This generalizes the effacement theorem of Quillen, Bloch-Ogus, and Gabber. As an application we obtain a new proof of the Gersten conjecture in this case. In fact, our proof yields a stronger version of the Gersten conjecture, which answers in particular questions of Colliot-Thélène—Hoobler—Kahn. We will also discuss applications that go beyond the original framework of the Gersten conjecture.

Domenico Valloni Reduction mod $$p$$ of the Noether problem

Let $$k$$ be any field and let $$V$$ be a linear and faithful representation of a finite group $$G$$. The Noether problem asks whether $$V/G$$ is a (stably) rational variety over k. It is known that if $$p = char(k) > 0$$ and $$G$$ is a $$p$$-group, then $$V/G$$ is always rational. On the other hand, Saltman and later Bogomolov constructed many examples of $$p$$-groups $$G$$ such that $$V/G$$ is not stably rational over the complex numbers. In this talk we study what happens over dvr of mixed characteristic $$(0,p)$$. We show that for all the examples found by Saltaman and Bogomologov, there cannot exist a smooth projective scheme over $$R$$ whose special and generic fibre are stably birational to $$V/G$$. In particular, this proves that $$P^n_R/G$$ never admits a relative resolution of singularities over $$R$$. The proof combines integral $$p$$-adic Hodge theory with the study of indefinitely closed differential forms in positive characteristic.

Claire Voisin Cycle classes on abelian varieties and the geometry of the Abel-Jacobi map

We discuss two properties of an abelian variety, namely, being a direct summand in a product of Jacobians and the weaker property of being "split". We relate the first property to the integral Hodge conjecture for curve classes on abelian varieties. We also relate both properties to the existence problem for universal zero-cycles on Brauer-Severi varieties over abelian varieties. A similar relation is established for the existence problem of a universal codimension 2 cycle on a cubic threefold.

### Location

Buildings: Erwin Schrödinger-Zentrum and the Johann von Neumann-Haus (where the math institute locates). See the schedule above for the individual rooms.

Erwin Schrödinger-Zentrum
Rudower Chaussee 26
12489 Berlin

Institut für Mathematik
Rudower Chaussee 25
Johann von Neumann-Haus
12489 Berlin

Both buildings are near each other in the Adlershof campus of the Humboldt University of Berlin. From the city center, take the south/airport-bound S-Bahn trains 8, 9, 45, 46 or 85 to the S-Adlershof station. Then take the west-bound trams (any) or buses 162, 163, 164 for 2 stops to Magnusstraße (Magnusstr.), or walk for about 10 minutes.

A detailed local map

### Acknowledgement

The event is generously supported by: