Humboldt-Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät
Institut für Mathematik

## Forschungsseminar "Algebraische Geometrie"

Sommersemester 2022

Time: Wednesday 13:15 - 14:45

Room: Online (Zoom) or see individual talks below for rooms.

TimeRoomSpeaker
04.05.2022 Room 3.007, Rudower Chaussee 25 Ruijie Yang (HU Berlin)
Title: Singularities of theta divisors of hyperelliptic curves
Abstract: It is a classical result that the theta divisor on the Jacobian variety associated to a smooth projective hyperelliptic curve has maximal dimension of singularities among indecomposable principally polarized abelian varieties. A version of the Schottky problem asks if this condition on the dimension of singularities characterizes hyperelliptic Jacobians.
Motivated by this problem, it is natural to study the singularities of hyperelliptic theta divisors in more details to understand why they are special. In this talk, I will explain a natural and explicit embedded resolution of hyperelliptic theta divisors inside Jacobians by successively blowing up (proper transforms of) Brill-Noether subvarieties. A key observation is that we can use the geometry of Abel-Jacobi maps and secant varieties of rational normal curves to avoid the analysis of singularities in the blow-up process. From the point of view of singularities of pairs, we obtain all the essential information.
If time permits, I will discuss how this resolution can be used to understand the mixed Hodge module structure on the vanishing cycle of a hyperelliptic theta divisor, using a global version of Esnault-Viehweg’s cyclic covering construction, limiting mixed Hodge structures on normal crossing divisors, and twisted D-modules. This is joint work (partially in progress) with Christian Schnell.
11.05.2022 3.007 Andrei Bud (HU Berlin)
Title: The Prym-Brill-Noether divisor
Abstract: Understanding the birational geometry of the moduli space $$\mathcal{R}_g$$ parametrizing Prym curves has been the subject of several papers, with great insight into this problem coming from the work of Farkas and Verra. Of particular importance for this study is finding divisors of small slope on the space $$\overline{\mathcal{R}}_g$$. Drawing parallels with the situation on $$\overline{\mathcal{M}}_g$$, we consider the Prym-Brill-Noether divisor and compute (some relevant coefficients of) its class. We will highlight the role of strongly Brill-Noether loci in understanding Prym-Brill-Noether loci. A consequence of our study is that the space $$\mathcal{R}_{14,2}$$ parametrizing $$2$$-branched Prym curves of genus $$14$$ is of general type.
18.05.2022 No Seminar
25.05.2022 1.115 Reinier Kramer (University of Alberta)
Title: The spin Gromov-Witten/Hurwitz correspondence
Abstract: There are two important ways to calculate the number of maps between Riemann surfaces with given conditions: Hurwitz theory is over a hundred years old and uses the monodromy representation to transport the problem to symmetric groups, representation theory, and the Kadomstev-Petviashvili (KP) integrable hierarchy. Gromov-Witten theory for curves recasts the problem as intersection theory on the moduli space of such (stable) maps. By work of Okounkov-Pandharipande, there is a strict correspondence between the two. Both of these sides have an analogue with spin: this takes into account a bundle on the source which squares to the canonical bundle. On the Hurwitz side, this has relations to spin-symmetric or Sergeev groups, and the BKP hierarchy, while on the Gromov-Witten side, these invariants arise from localising the invariants of surfaces with smooth canonical divisor. I will explain that, at least for P^1, there is a correspondence between these two sides as well, which hinges on a spin analogue of the Ekedahl-Lando-Shapiro-Vainshtein formula and cosection localisation. This is joint work (partially in progress) with Alessandro Giacchetto, Danilo Lewański, and Adrien Sauvaget.
25.05.2022, 14:45 1.115 Philippe Eyssidieux (Grenoble)
Title: $$L_2$$ Cohomology for Hodge Modules
Abstract: The talk will be based on arxiv:2203.06950[math.AG]. For an infinite Galois covering space of a compact Kähler manifold, we define $$L_p$$ cohomology for $$1\le p<+\infty$$ for various types of coefficients (perverse sheaves, coherent D-modules, Mixed Hodge Modules) and explain how to control it when $$p=2$$. The formalism encompasses both my 2000 work on $$L_2$$-coherent cohomology (see also the independant and simultaneous work of Campana-Demailly) and Dingoyan's 2013 work on $$L_2$$-De Rham cohomology of an open subset. We describe a conjectural MHS on the reduced $$L_2$$-cohomology of a MHM and explain what can be proved with today's technology.
01.06.2022, 16:15 Zoom Ziquan Zhuang (MIT)
Title: Finite generation and Kähler-Ricci soliton degenerations of Fano varieties
Abstract: By the Hamilton-Tian conjecture on the limit behavior of Kähler-Ricci flows, every complex Fano manifold degenerates to a Fano variety that has a Kähler-Ricci soliton. In this talk, I'll discuss the algebro-geometric analogue of this statement and explain its connection to certain finite generation results in birational geometry. Based on joint work with Harold Blum, Yuchen Liu and Chenyang Xu.
08.06.2022, 16:15 Zoom Nicola Tarasca (Virginia Commonwealth University)
Title: Incident varieties of algebraic curves and canonical divisors
Abstract: The theory of canonical divisors on curves has witnessed an explosion of interest in recent years, motivated by recent developments in the study of limits of canonical divisors on nodal curves. Imposing conditions on canonical divisors allows one to construct natural geometric subvarieties of moduli spaces of pointed curves, called strata of canonical divisors. These strata are the projection on moduli spaces of curves of incidence varieties in the projectivized Hodge bundle. I will present a formula for the class of such incident varieties over the locus of pointed curves with rational tails. The formula is expressed as a linear combination of tautological classes indexed by decorated stable graphs, with coefficients enumerating appropriate weightings. I will conclude discussing applications to the study of relations in the tautological ring. Joint work with Iulia Gheorghita.
15.06.2022 No Seminar
22.06.2022 3.007 Carl Lian (HU Berlin)
Title: Degenerations of complete collineations and geometric Tevelev degrees of $$\mathbb{P}^r$$
Abstract: This is a report on work in progress. We will discuss a complete answer, in terms of Schubert calculus, to the problem of enumerating maps of degree $$d$$ from a fixed general curve of genus $$g$$ to $$\mathbb{P}^r$$ satisfying incidence conditions at the appropriate number of marked points, that is, we compute the geometric Tevelev degrees of $$\mathbb{P}^r$$. Previously, after the work of many people, the answers were known only when $$r = 1$$, or when d is large compared to $$r$$, $$g$$; in the latter case, the answers agree with virtual counts in Gromov-Witten theory, but when $$d$$ is small, the situation is considerably more subtle. The method proceeds by reduction to genus 0 via limit linear series, and then by an analysis of certain Schubert-type cycles on moduli spaces of complete collineations upon further degeneration.
29.06.2022, 16:15 Zoom Junliang Shen (Yale)
Title: A tale of three moduli spaces of sheaves
Abstract: I will discuss cohomological structures for three moduli spaces of sheaves: the moduli of vector bundles on a curve, the moduli of Higgs bundles on a curve, and the moduli of 1-dimensional torsion sheaves on $$\mathbb{P}^2$$. These moduli spaces have been studied intensively from various perspectives. In recent years, enumerative geometry and string theory sheds new lights on the cohomological structure of these classical moduli spaces. In the talk I will discuss some results and conjectures in this direction; this concerns the $$\chi$$-independence phenomenon, tautological generators, and the $$P=W$$ conjecture. Based on joint works with Davesh Maulik, Weite Pi, and an on-going project with Yakov Kononov and Weite Pi.
06.07.2022, 16:15 Zoom Xiaolei Zhao (UCSB)
Title: Gushel-Mukai varieties, stability conditions, and moduli of stable objects
Abstract: Gushel-Mukai varieties are smooth Fano varieties of dimension between 3 to 6, Picard rank 1, degree 10 and co-index 3. Depend on the parity of the dimension, their derived categories contain a so called Kuznetsov component, behaving similarly to an Enriques surface or a K3 surface. I will survey the construction of Bridgeland stability conditions on the Kuznetsov component, and some properties about moduli of stable objects on it. Applications to the geometry of Fano varieties will also be explained. This is based on joint work with Alex Perry and Laura Pertusi.
13.07.2022
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20.07.2022 Zoom Haoyang Guo (Max Planck Institute for Mathematics)
Title: Prismatic approach to p-adic local systems
Abstract: Let X be a smooth proper scheme over a p-adic field that admits a good reduction. Inspired by the de Rham comparison theorem in complex geometry, Grothendieck asked if there is a "mysterious functor", relating étale cohomology of the generic fiber and crystalline cohomology of the special fiber. The question was subsequently answered by Fontaine, Faltings and many others' work, which was one of the foundational results in the p-adic Hodge theory. In particular, this motivates the definition of a p-adic local system being crystalline, generalizing the representational property of the etale cohomology of X as above. In this talk, we will give an overview for crystalline representations and crystalline local systems. Building on the recent advance of Bhatt-Scholze, we then introduce the prismatic approach to crystalline local systems. This is a joint work with Emanuel Reinecke.
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