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

Online Forschungsseminar "Algebraische Geometrie"

Sommersemester 2021

Das Forschungsseminar findet mittwochs in der Zeit von 16:00 - 17:00 Uhr in Zoom statt.

Seminar: Algebraic Geometry an der FU

14.04.2021 Andrey Soldatenkov (Humboldt-Universität zu Berlin)
Title: Holonomy of the Obata connection on hypercomplex manifolds

Abstract: The algebra of quaternions has been a focus of attention in many branches of mathematics ever since its introduction by Hamilton. One may think that quaternions form a noncommutative finite extension of the field of complex numbers. For a geometer, it is natural to wonder if there exists a suitable notion of a quaternionic variety, analogous to a complex algebraic variety. I will try to give an introduction to this circle of ideas, explain how one can approach quaternionic (or hypercomplex) geometry and what natural problems arise in this context. One important notion in hypercomplex geometry is the Obata connection, the unique torsion-free connection that preserves the action of the quaternions. I will present some results on the study of its holonomy.
21.04.2021 Alessandro Verra (Universita Roma Tre)
Title: The Igusa quartic and the Prym map

Abstract: The Igusa, or the Castelnuovo-Richmond quartic is a famous hypersurface of the complex projective 4-space known for its ubiquity in algebraic geometry. It is related to the Prym map \(P\) in genus 6. As is well known the map \(P\) has degree 27 and dominates the moduli space of 5-dimensional principally polarized abelian varieties. Other maps with the same monodromy are associated to \(P\) and reflect related configurations. Among these these of particular importance is the map \(J: D \rightarrow A_5\), with fibre the configuration of double sixes of lines of the cubic surface. We describe \(J\) geometrically, showing that it is birationally equivalent to the period map for the moduli space \(D\) of 30-nodal quartic threefolds, cutting twice a quadratic section of the Igusa quartic.
28.04.2021 (start at 17:00) Hannah Larson (Stanford University)
Title: The rational Chow rings of \(M_7\), \(M_8\), and \(M_9\)

Abstract: The rational Chow ring of the moduli space \(M_g\) of curves of genus \(g\) is known for \(g \leq 6\). In each of these cases, the Chow ring is tautological (generated by certain natural classes known as kappa classes). In recent joint work with Sam Canning, we prove that the rational Chow ring of \(M_g\) is tautological for \(g = 7, 8, 9\), thereby determining the Chow rings by work of Faber. In this talk, I will give an overview of our approach, with particular focus on the locus of tetragonal curves (special curves admitting a degree 4 map to \(\mathbb{P}^1\)).
05.05.2021 François Greer (Stony Brook University)
Title: A tale of two Severi curves

Abstract: Let \((S,L) \) be a general polarized K3 surface of degree \(2g-2 \). A general member of the linear system \(|L|\simeq \mathbb P^g \) is a smooth curve of genus \(g \). For \(0\leq h\leq g \), define the Severi variety \(V_h(S,L)\subset |L| \) to be the locus of curves with geometric genus \(\leq h \). As expected, \(V_h(S,L) \) has dimension \(h \). We consider the case \(h=1 \), where the Severi variety is a (singular) curve. Our first result is that the geometric genus of \(V_1(S,L) \) goes to infinity with \(g \); we give a lower bound \(\sim e^{c\sqrt{g}} \). Next we consider the analogous question for Severi curves of a rational elliptic surface, and give a polynomial upper bound instead. Modular forms play a central role in both arguments.
12.05.2021 (17:00) Mina Aganagic (University of California Berkeley)
Title: Khovanov Homology from Mirror Symmetry

Abstract: Khovanov showed, more than 20 years ago, that there is a deeper theory underlying the Jones polynomial. The “knot categorification problem” is to find a uniform description of this theory, for all gauge groups, which originates from physics, or geometry. I will describe two solutions to this problem, which I recently discovered, related by a version of two dimensional (homological) mirror symmetry. The theories are significantly more efficient than the algebraic descriptions mathematicians have found, even in the Khovanov homology case.
19.05.2021 (17:00) Salim Tayou (Harvard University)
Title: Equidistribution of Hodge loci (notes)

Abstract: Given a polarized variation of Hodge structures, it is a classical result that the Hodge locus is a countable union of proper algebraic subvarieties. In this talk, I will explain a general equidistribution theorem for these Hodge loci and explain several applications: equidistribution of higher codimension Noether-Lefschetz loci, equidistribution of Hecke translates of a curve in \(A_g\) and equidistribution of some families of CM points in Shimura varieties. The results of this talk are joint work with Nicolas Tholozan.
26.05.2021 Rahul Pandharipande (ETH Zürich)
Title: Tevelev degrees and Hurwitz moduli spaces

Abstract: I will explain various numerical and cohomological questions related to Hurwitz moduli spaces (including older results with Faber on tautological classes and newer calculations with Cela and Schmitt on Tevelev degrees).
02.06.2021 David Holmes (University of Leiden)
Title: The double-double ramification cycle

Abstract: A basic question in the geometry of Riemann surfaces is to decide when a given divisor of degree 0 is the divisor of a rational function (is principal). In the 19th century Abel and Jacobi gave a beautiful solution: one writes the divisor as the boundary of a 1-cycle, and the divisor is principal if and only if every holomorphic differential integrates to zero against this cycle. From a modern perspective it is natural to allow the curve and divisor to vary in a family, perhaps allowing the curve to degenerate to a singular (stable) curve so that the corresponding moduli space is compact. The double ramification cycle can then be seen as a virtual fundamental class of the locus in the moduli space of curves over which our divisor becomes principal. We will focus on two basic questions: where does the double ramification cycle naturally live, and what happens when we intersect two double ramification cycles? We will see why (logarithmically) blowing up the moduli space can make life easier. This is joint work with Rosa Schwarz, building on earlier joint work with Aaron Pixton and Johannes Schmitt.
09.06.2021 Andrea di Lorenzo (Humboldt-Universität zu Berlin)
Title: The integral Chow ring of the stack of stable 1-pointed curves of genus two

Abstract: Moduli stacks of curves play a prominent role in algebraic geometry. In particular, their rational Chow rings have been the subject of intensive research in the last forty years, since Mumford first investigated the subject. There is also a well defined notion of integral Chow ring for these stacks: this is more refined, but also much harder to compute. In this talk I will present the computation of the integral Chow ring of the stack of stable 1-pointed curves of genus two, obtained by using a new approach to this type of questions (joint project with Michele Pernice and Angelo Vistoli).
16.06.2021 Gerard Freixas i Montplet (IMJ-PRG)
Title: Complex Chern-Simons and the first tautological class (slides)

Abstract: In this talk I will propose a construction of the complex Chern-Simons line bundle, in the context of a family of compact Riemann surfaces and a relative moduli space of flat vector bundles on it. The construction is inspired by Deligne's functorial interpretation of Arakelov geometry, where direct images of characteristic classes of hermitian vector bundles are lifted to the level of hermitian line bundles. In our setting, hermitian metrics are replaced by flat relative connections, and non-abelian Hodge theory is a fundamental tool in the approach. We will discuss some properties of the complex Chern-Simons line bundle, and an application to a differential geometric incarnation of the first tautological class on the moduli space of curves. This is joint work with Dennis Eriksson and Richard Wentworth.
23.06.2021 No seminar

30.06.2021 Rahul Pandharipande (ETH Zürich)
Title: Tevelev degrees and Hurwitz moduli spaces (Part 2)

07.07.2021 Philip Engel (UGA)
Title: Compact K3 moduli

Abstract: This is joint work with Valery Alexeev. By the Torelli theorem, the moduli space \(F_g\) of polarized K3 surfaces is the quotient of a 19-dimensional Hermitian symmetric space by the action of an arithmetic group. In this capacity, it admits a natural class of "semitoroidal compactifications," built from periodic tilings of 18-dimensional hyperbolic space. On the other hand, \(F_g\) also admits "stable pair compactifications": Choosing canonically on any polarized K3 surface \(X\) an ample divisor \(R\), there is a compact moduli space of "stable pairs" containing the K3 pairs \((X,R)\) as an opensubset.
I will discuss two theorems in the talk: First, there is a simple criterion on \(R\), "recognizability", which implies that the normalization of a stable pair compactification is semitoroidal. Second, the sum of geometric genus zero curves in the polarization is recognizable. This gives rise to a modular semitoroidal compactification for all \(g\).


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