# NoGAGS 2018: North German Algebraic Geometry Seminar 2018

## Abstracts for the talks

**Dominic Bunnett ***Freie Universität Berlin*

Time: Thursday, November 08, 2018, 16:00

**Moduli of hypersurfaces in weighted projective space **

The moduli space of smooth hypersurfaces in projective space was constructed by Mumford in the 60’s using his newly developed classical (a.k.a. reductive) Geometric Invariant Theory. I wish to generalise this construction to hypersurfaces in weighted projective space (or more generally orbifold toric varieties). The automorphism group of a toric variety is in general non-reductive and I will use new results in non-reductive GIT, developed by F. Kirwan et al., to construct a moduli space of quasismooth hypersurfaces. I will give geometric characterisations of notions of stability arising from non-reductive GIT.

**Marco D'Addezio ***Freie Universität Berlin*

Time: Friday, November 09, 2018, 14:15

**On the semi-simplicity conjecture for Qab**

We survey some old and new developments of the Grothendieck-Serre-Tate conjectures, with a particular attention to the semi-simplicity conjecture. We recall the role of Deligne's theory of weights for the semi-simplicity conjecture in positive characteristic. Then we switch to characteristic 0 and we explain some analogies with the previous situation.

**Rahul Pandharipande ***ETH Zürich*

Time: Thursday, November 08, 2018, 13:00

**Enumerative geometry and the holomorphic anomaly equation**

I will give a (gentle) introduction to the role of the holomorphic anomaly equation in curve counting problems starting with maps to elliptic curves, proceeding to local toric geometries, and concluding with the famous quintic 3-fold. Though I will discuss joint work with Hyenho Lho, much of what I will say is work of others (Pixton, Oberdieck-Pixton, Chen-Guo-Janda-Ruan, Jun Li and collaborators).

**Sam Payne ***University of Texas at Austin*

Time: Thursday, November 08, 2018, 17:15

**Top weight cohomology of M_g**

I will discuss the topology of a space of stable tropical curves of genus g with volume 1, whose reduced rational homology is canonically identiﬁed with the top weight cohomology of M˙g and also with the homology of Kontsevich’s graph complex. As one application, we show that Hˆ4g-6(M˙g) is nonzero for inﬁnitely many g. This disproves a recent conjecture of Church, Farb, and Putman as well as an older, more general conjecture of Kontsevich. We also give an independent proof of a recent theorem of Willwacher, that homology of the graph complex vanishes in negative degrees, using the identiﬁcations above and known vanishing results for M˙g. Joint work with M. Chan and S. Galatius.

**Jason van Zelm ***Humboldt-Universität zu Berlin*

Time: Friday, November 09, 2018, 09:15

**Computing classes of admissible covers**

Let Adm(g,h,d) be the space of degree d admissible covers C → D of a genus h curve D by genus g curves C. There is a natural map f : Adm(g,h,d) → Mgnbar into the moduli space of stable curves taking the source curve of an admissible cover and forgetting everything else. When the class [f(Adm(g,h,d))] is tautological we can try to express this class in terms of a known basis for the tautological ring of Mgnbar. We will discuss several strategies for making these computations and give a number of examples.

**Milena Wrobel ***MPI MiS, Leipzig*

Time: Thursday, November 08, 2018, 14:30

**The anticanonical complex - a combinatorial tool for Fano varieties**

Toric Fano varieties are in one to one correspondence with certain lattice polytopes, the so called Fano polytopes. Moreover classification of toric Fano varieties with respect to their singularity type turns out to be purely combinatorial: the position of lattice points in the Fano polytope determines the singularity type. The anticanonical complex has been introduced as a natural generalisation of the toric Fano polytope and so far has been succesfully used for the study of varieties with a torus action of complexity one. We enlarge the area of application of the anticanonical complex to Fano varieties with torus action of arbitrary complexity, for example arrangement varieties. In particular, we show that the possibility to apply the anticanonical complex to these varieties is connected to certain properties of their quotients.

**Fei Xie ***Bielefeld University*

Time: Friday, November 09, 2018, 12:45

**Derived categories of fibrations of quintic del Pezzo surfaces**

I will provide two approaches for fining a semiorthogonal decomposition of the derived category of fibrations of quintic del Pezzo surfaces with RDP singularities. Similar to Kuznetsov’s work on sextic del Pezzo surfaces, the components of the semiorthogonal decomposition can be interpreted as the moduli spaces of semistable sheaves on fibers with fixed Hilbert polynomials. Alternatively, there is a rank 2 vector bundle on a quintic del Pezzo surface with RDP singularities that embeds the surface as a linear section of a Grassmannian. The semiorthogonal decomposition can be obtained by applying Homological Projective Duality. This is work in progress.

**Ziyu Zhang ***Leibniz Universität Hannover*

Time: Friday, November 09, 2018, 10:45

**Formality conjecture and moduli spaces of sheaves on K3 surfaces**

The formality conjecture for K3 surfaces, formulated by D.Kaledin and M.Lehn, states that on a complex projective K3 surface, the differential graded algebra RHom(F,F) is formal for any coherent sheaf F polystable with respect to an ample line bundle. In this talk, I will explain how to combine techniques from twistor spaces, dg categories and Fourier-Mukai transforms to prove this conjecture, and how to generalize it to derived objects. Based on joint work with Nero Budur.

## Date and Location

**November 08 - 09, 2018**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**Mateusz Michalek**

MPI für Mathematik in den Naturwissenschaften

## Administrative Contact

**Saskia Gutzschebauch**

MPI für Mathematik in den Naturwissenschaften

Contact by Email