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

Forschungsseminar "Algebraische Geometrie"

Wintersemester 2023


Time: Wednesday 13:15 - 14:45

Room: 3.007 John von Neumann-Haus

Humboldt Arithmetic Geometry Seminar

Seminar: Algebraic Geometry an der FU



TimeRoomSpeaker
28.09.2023
Hilbert schemes of points
04.10.2023, 13:15 Rahul Pandharipande (ETH Zürich)
Title: The Gromov-Witten/Donaldson-Thomas correspondence, Hilbert schemes of the affine plane and the moduli of abelian varieties
Abstract: I will explain how these three directions of study are fundamentally linked.
30.10.2023-31.10.2023
Hodge theory, tropical geometry and o-minimality
01.11.2023 Kristian Ranestad (University of Oslo)
Title: Quaternary quartic forms and Gorenstein rings
Abstract: The Betti tables of their apolar rings give rise to a stratification of the space of quartic forms. The strata may be characterized by possible power sum decompositions and liftings to Calabi-Yau 3-folds. I shall report on work with G. And M. Kapustka, H. Schenk, M. Stillman and B. Yuan.
08.11.2023 Alessandro Verra (Università Roma Tre)
Title: From Enriques surfaces to the Artin-Mumford counterexample
Abstract: The talk deals with the multiple relations between Enriques surfaces and rationality problems. Artin-Mumford's counterexample to Lueroth's problem is revisited: the role of Enriques surfaces, the family of Reye congruences is emphasized and the 2-torsion cohomology of the threefold is geometrically reconstructed from that of these surfaces. The same construction extends to higher dimensions.
15.11.2023 Frank-Olaf Schreyer (Universität des Saarlandes)
Title: Tate resolutions of Gorenstein Rings and a construction from Clifford modules of complete intersection of two quadrics
Abstract: The concept of MCM approximations of Auslander-Buchweitz is a beautiful concept which builds on Tate resolution. I will decribe the complexes explicitly in case for the case of the coordinate ring a complete intersection \((x_1,\ldots,x_n)\) as a module over a coordinate ring of a further complete intersection \((q_1,\ldots,q_r)\).
In the second part I will explain how one can directly construct Tate resolution from a module over the Clifford algebra of a complete intersection of two quadrics \(X \subset \mathbb{P}^{2g+1}\) and their relation to Ulrich bundles on \(X\).
29.11.2023 Giancarlo Urzúa (Pontificia Universidad Católica de Chile)
Title: The birational geometry of Markov numbers
Abstract: The projective plane is rigid. However, it may degenerate to surfaces with quotient singularities. After the work of Bădescu and Manetti, Hacking and Prokhorov 2010 classified these degenerations completely. They are \(\mathbb{Q}\)-Gorenstein partial smoothings of \(\mathbb{P}(a^2,b^2,c^2)\), where \(a,b,c\) satisfy the Markov equation \(x^2+y^2+z^2=3xyz\). Let us call them Markovian planes. They are part of a bigger picture of degenerations with Wahl singularities, where there is an explicit MMP whose final results are either \(K\) nef, smooth deformations of ruled surfaces, or Markovian planes. Although it is a final result of MMP, we can nevertheless run MMP on small modifications of Markovian planes to obtain new numerical/combinatorial data for Markov numbers via birational geometry. New connections with Markov conjecture (i.e. Frobenius Uniqueness Conjecture) are byproducts. This is joint work with Juan Pablo Zúñiga (Ph.D. student at UC Chile), the pre-print can be found here.
THURSDAY 07.12.2023, 14h Igor Burban (Universität Padelborn)
Title: Algebraic geometry of the torus model of the fractional quantum Hall effect
Abstract: The experimental discovery of the quantum Hall effect is widely considered to be a one of the major events in the condensed matter physics in the second half of the twentieth century. Both experimental and theoretical aspects of this phenomenon still continue to attract an enormous attention.
In 1993 Keski-Vakkuri and Wen introduced a model for the quantum Hall effect based on multilayer two-dimensional electron systems satisfying quasi-periodic boundary conditions. Such a model is specified by a choice of a complex torus \(E\) and a symmetric positively definite matrix \(K\) of size \(g\) with integer coefficients.
The space of the corresponding wave functions turns out to be \(d\)-dimensional, where \(d\) is the determinant of \(K\). I am going to explain a construction of a hermitian holomorphic bundle of rank \(d\) on the abelian variety \(A\) (which is the \(g\)-fold product of the torus \(E\) with itself), whose fibres can be identified with the space of wave function of Keski-Vakkuri and Wen. A rigorous construction of this "magnetic bundle" involves the technique of Fourier-Mukai transforms on abelian varieties. This bundle turns out to be simple and semi-homogeneous. Moreover, for special classes of the matrix \(K\), the canonical Chern-Weil connection of the magnetic bundle is shown to be projectively flat.
This talk is based on a joint work with Semyon Klevtsov (arXiv:2309.04866).
13.12.2023 Gergely Berczi (Aarhus University)
Title: Geometry of the Hilbert scheme of points on manifolds, part II
Abstract: While the Hilbert scheme of points on surfaces is well-explored and extensively studied, the Hilbert scheme of points on higher-dimensional manifolds proves to be exceptionally intricate, exhibiting wild and pathological characteristics. Our understanding of their topology and geometry is very limited.
In these talks I will present recent results on various aspects of their geometry. I will discuss
i) the distribution of torus fixed points within the components of the Hilbert scheme (work with Jonas Svendsen)
ii) intersection theory of Hilbert schemes: a new formula for tautological integrals over geometric components, and applications in enumerative geometry
iii) new link invariants of plane curve (and hypersurface) singularities coming from p-adic integration and Igusa zeta functions (work with Ilaria Rossinelli).
This talk will be relatively independent from part I on 12th December at the Arithmetic Geometry Seminar
20.12.2023, 13:15-14:15 Simon Felten (Columbia University)
Title: Global logarithmic deformation theory
Abstract: A well-known problem in algebraic geometry is to construct smooth projective Calabi-Yau varieties \(X\). In the smoothing approach, one constructs first a degenerate (reducible) Calabi-Yau variety \(V\) by gluing pieces. Then one aims to find a family with special fiber \(V\) and smooth general fiber \(X\). In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber \(V\). This is achieved via the logarithmic Bogomolov-Tian-Todorov theorem as well as its variant for pairs of a log Calabi-Yau space \(f_0: X_0 \to S_0\) and a line bundle \(\mathcal{L}_0\) on \(X_0\). Both theorems are a consequence of the abstract unobstructedness theorem for curved Batalin-Vilkovisky algebras.
20.12.2023, 14:45-15:45 Hélène Esnault (FU Berlin/Harvard/Copenhagen)
Title: Survey on some arithmetic properties of rigid local systems
Abstract: A central conjecture of Simpson predicts that complex rigid local systems on a smooth complex variety come from geometry. In the last couple of years, we proved some arithmetic consequences of it: integrality (using the arithmetic Langlands program), F-isocrystal properties, crystallinity of the underlying p-adic representation (using the Cartier operator over the Witt vectors and the Higgs-de Rham flow) (for Shimura varieties of real rank at least 2, this is the corner piece of Pila-Shankar-Tsimerman's proof of the André-Oort conjecture), weak integrality of the character variety (using de Jong's conjecture proved with the geometric Langlands program) (yielding a new obstruction for a finitely presented group to be the topological fundamental group of a smooth complex variety).
We'll survey some aspects of this (please ask if there is something on which you would like me to focus on). The talk is based mostly on joint work with Michael Groechenig, also, even if less, with Johan de Jong.
10.01.2024 Thomas Walpuski (HU Berlin)
Title: The Gopakumar–Vafa finiteness conjecture
Abstract: The purpose of this talk is to illustrate an application of the powerful machinery of geometric measure theory to a conjecture in Gromov–Witten theory arising from physics. Very roughly speaking, the Gromov–Witten invariants of a symplectic manifold \((X,\omega)\) equipped with a tamed almost complex structure \(J\) are obtained by counting pseudo-holomorphic maps from mildly singular Riemann surfaces into \((X,J)\). It turns out that Gromov–Witten invariants are quite complicated (or “have a rich internal structure”). This is true especially for if \((X,\omega)\) is a symplectic Calabi–Yau 3–fold (that is: \(\mathrm{dim}X=6\), \(c_1(X,\omega) = 0\)).
In 1998, using arguments from M–theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi–Yau 3–folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov–Witten invariants by a transformation of the generating series. The Gopakumar–Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition.
The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that Gromov’s compactness theorem can (and should!) be replaced with the work of Federer–Flemming, Allard, and De Lellis–Spadaro–Spolaor. This upgrade of Ionel and Parker’s cluster formalism proves both the integrality and finiteness conjecture.
This talk is based on joint work with Eleny Ionel and Aleksander Doan.
THURSDAY 11.01.2024, 13.15h Andreas Kretschmer (Magdeburg)
Title: Characteristic Numbers for Cubic Hypersurfaces
Abstract: Given an \(N\)-dimensional family \(F\) of subvarieties of some projective space, the number of members of \(F\) tangent to \(N\) general linear spaces is called a characteristic number for \(F\). More general contact problems can often be reduced to the computation of these characteristic numbers. However, determining the characteristic numbers themselves can be very difficult, and very little is known even for the family of smooth degree \(d\) hypersurfaces of projective \(n\)-space as soon as both \(n\) and \(d\) are greater than 2. In this talk I will survey the construction of a so-called 1-complete variety of cubic hypersurfaces, generalizing Aluffi's variety of complete plane cubic curves to higher dimensions. This allows us to explicitly compute, for instance, the numbers of smooth cubic surfaces, threefolds and fourfolds tangent to 19, 34 and 55 lines, respectively.
24.01.2024 Gian Pietro Pirola (Università di Pavia)
Title: Second fundamental form on \(\mathcal{M}_g\) associated to the period map and its asymptotic lines
Abstract: The aim is to study the second fundamental form associated with the image period map for curves. We present some computational improvements that allow classifying the asymptotic low-rank complex line with respect to the infinitesimal variation of the Hodge structure map and its relation to the Clifford index. This is a joint work with Elisabetta Colombo and Paola Frediani.
14.02.2024, 13:15-14:15 Andrei Bud (Goethe Universität Frankfurt)
Title: Degenerations of Prym-Brill-Noether loci
Abstract: I will describe the Prym-Brill-Noether loci for curves in the boundary of the moduli of Prym curves. As consequences of this, I prove the irreducibility of the Universal Prym-Brill-Noether locus and compute the class of the Prym-Brill-Noether divisor.
14.02.2024, 14:30-15:30 Rahul Pandharipande (ETH Zürich)
Title: Tautological projections and the cohomology of the moduli space of abelian varieties
Abstract: I will construct the projection operator on the Chow ring for the moduli of abelian varieties and compute many new examples (related to the geometry of the Lagrangian Grassmannian) elucidating the structure of this ring.
THURSDAY 22.02.2024 A day of Algebraic Geometry (Room 1.115)
10:00-11:00 Annette Werner (Frankfurt): Non-abelian p-adic Hodge theory
Abstract: In this talk I will explain background, achievements and challenges in the quest to find non-archimedean versions of Simpson's celebrated correspondence, which on Kähler manifolds relates representations of the fundamental group to certain Higgs bundles. In the non-archimedean world, vector bundles for the v-topology on Scholze's diamonds have proven to be a useful framework to study analogous problems.
After explaining these tools, I will present an analog of the classical correspondence on abeloid varieties.
11:30-12:30 Bernd Sturmfels (Leipzig): Kinematics on \(\mathcal{M}_{0,n}\)
Abstract: Minimal kinematics identifies likelihood degenerations where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov-Huh. We characterize all choices of minimal kinematics on the moduli space \(\mathcal{M}_{0,n}\). These choices are motivated by the CHY model in physics and they are represented combinatorially by 2-trees. We compute 2-tree amplitudes, and we explore extensions to non-planar on-shell diagrams, here identified with the hypertrees of Castravet-Tevelev.
14:00-15:00 Raluca Vlad (Leipzig): Wonderful polytopes
Abstract: A positive geometry consists of a real projective variety and a semialgebraic subset (its “positive part”), together with a canonical rational form which satisfies a recursive definition when restricted to the boundary of the semialgebraic set. Positive geometries have been objects of interest in physics, and have recently started being explored mathematically. In my talk, I will focus on hyperplane arrangements in projective space. Regions in a hyperplane arrangement complement are polytopes, which are known to be positive geometries. I will discuss when such a region remains a positive geometry after taking the wonderful compactification of the arrangement. This talk is based on work in progress with S. Brauner, C. Eur, and L. Pratt.
15:30-16:30 Diane Maclagan (Warwick): Tropical vector bundles
Abstract: Tropicalization replaces a variety by a combinatorial shadow that preserves some of its invariants. When the variety is a subspace of projective space the tropical variety is determined by a (valuated) matroid. I will review this, and discuss a resulting definition for a tropical vector bundle in the context of tropical scheme theory. This is joint work with Bivas Khan.
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