Springe direkt zu Inhalt

# Seminar Algebraische Geometrie

Winter Semester 2022/2023

talks will be announced here in advance

Summer Semester 2022

Unless otherwise specified talks will be hybrid (webEx and live) at 16:15h in Room 6 (Königin-Luise-Straße 24-26)

https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m210c477e0a4efb8dc2dc2d0d50b81694
Meeting number: 2730 277 0429, Password: cY28qxt6sSt

Schedule

04.07.2022

Christian Lehn (TU Chemnitz)

Titel: Singular varieties with trivial canonical class

Abstract: I will report on a recent joint work with Bakker and Guenancia building on work by many others on the classification of singular varieties with trivial canonical class. This includes the decomposition theorem which says that such a variety is up to a finite cover isomorphic to a product of a torus, irreducible Calabi-Yau (ICY) and irreducible symplectic varieties (ISV). The proof uses a reduction argument to the projective case which in turn is possible due to advances in deformation theory and a certain result about limits of Kähler Einstein metrics in locally trivial families.

27.06.2022

Lutz Hille (Universität Münster)

Titel: The Markov Equation and Polynomial Invariants for Triangulated Categories

Abstract: For any full exceptional sequence of vector bundles on the projective plane, the three ranks satisfy an equation  known as Markov equation: the sum of the squares of the three ranks minus  three times it’s product is zero. All solutions can be constructed recursively from the smallest solution (1,1,1) using the tree of mutations. The same equation also appears in many other situations, like for cluster algebras or matrices in Gl_2.

The principal aim of this talk is to generalize this equation to full exceptional sequences of larger length n. A polynomial invariant is a polynomial f in (n -1)n/2 variables x_i,j for i<j and I,j between 1 and n, so that f evaluated at the Euler characteristic <E_i, E_j> of any full exceptional sequence (E_1, … ,E_n) does not depend on the chosen exceptional sequence. In particular, it is an invariant of the triangulated category.

Depending on the length n of a full exceptional sequence we can ask about a maximal independent set of such invariants. Such an invariant is also invariant under the braid group action defined by mutations. We compare both notions and determine a complete set of polynomial invariants

13.06.2022

Francesca Cioffi (Università degli Studi di Napoli Federico II)

Titel: Liftings of projective schemes

Abstract: A classical approach to investigate a closed projective scheme X consists in considering a general hyperplane section of X, because many properties of X are inherited by a general hyperplane section and can be easier recognized in subschemes of lower dimension.

The inverse problem that consists in finding projective schemes $W$ having a given hyperplane section Y is called a lifting problem. Investigations in this topic can produce methods to obtain schemes  with specific properties.

I show that the liftings of Y with a given Hilbert polynomial can be parameterized by a scheme which is obtained by gluing suitable affine subschemes of a Hilbert scheme. Both constructive and classical techniques are used. The constructive methods are borrowed from Groebner and marked bases theory.

This talk originates from a joint paper with C. Bertone and D. Franco, "Functors of liftings of projective schemes", J. Symbolic Comput. 94  (2019), 105-125.

06.06.2022 No talk

30.05.2022 ONLY WEBEX!!

Andreas Hochenegger (Politecnico di Milano)

Titel: Relations among P-twists

Abstract: On a K3 surface, rational curves and line bundles give rise to  nteresting autoequivalences of its derived category, so-called spherical twists. It was shown by P. Seidel and R. Thomas, that these spherical twists are mirror-dual to Dehn twists in  symplectic geometry.

Moreover, they showed that given a chain of rational curves, the associated spherical twists satisfy braid relations. Generalising from K3 surfaces to hyperkähler varieties, D. Huyrbechts and R. Thomas showed that the corresponding generalisation of the spherical twists are P^n-twists.

In this talk, I will introduce these autoequivalences, and speak about the possible relations among them. This talk is about joint work in progress with Andreas Krug.

23.05.2022 ONLY WEBEX!!

Kiumars Kaveh (University of Pittsburgh)

Titel: An introduction to Arthur-Selberg trace formula and the polyhedral combinatorics involved

Abstract: We start by a brief and friendly introduction to Arthur-Selberg trace formula that vastly generalizes Poisson summation formula in Fourier analysis. It concerns the trace of left regular action of a reductive group G on the space $L^2(G/\Gamma)$ where $\Gamma$ is a discrete (arithmetic) subgroup.

The combinatorics involved is closely related to compactifications of ''locally symmetric spaces'' (which btw are hyperbolic manifolds). When $G/\Gamma$ is not compact the integral of the kernel of the regular action may not be convergent. To remedy this major difficulty, Arthur introduced his "truncation operator" which amounts to "truncating" the kernel function using certain polyhedral data resulting in an absolutely convergent integral.

We will introduce Arthur's truncation operation and discuss the connection with polyhedral/toric combinatorics. If time permits, we will mention a recent result (joint with Mahdi Asgari, Oklahoma State) that is an extraction of combinatorial content of Arthur's truncation and can be thought of Arthur's truncation operation on a toric variety.

16.05.2022

Rosa M. Miró-Roig (Universitat de Barcelona)

Titel: On the Existence of Ulrich Bundles on Smooth Surfaces

Abstract: In my talk I will discuss the existence of rank $r$ undecomposable Ulrich bundles on a general surface $S\subset \PP^3$ of degree $d$. More concretely, I will address the problem of characterizing the set $\{ (r,d)\in \NN \mid \exits a rank$rR Ulrich bundle on a general surface $S$ of degree $d$ in \PP^3 \}$. This problem is equivalent to the problem of determining whether an homogeneous form$F\in k[x_0,\cdots ,x_n]$of degree$d$(or one of its powers) can be written as the determinant of a matrix with linear entries. In my talk, I will summarize what it is known so far. 09.05.2022 Alexandru Constantinescu (Freie Universität Berlin) Titel: Cotangent Cohomology for Matroids Abstract: I will talk about the 1st and 2nd cotangent cohomology module of the Stanley-Reisner ring associated to a matroid. I will start with an upper bound for the dimensions of the multigraded components of T1 for arbitrary simplical complexes. For specific degrees, I will prove that these bounds are reached if and only if the simplicial complex is a matroid, obtaining thus a new characterization for matroids. Furthermore, the graded first cotangent cohomology turns out to be a complete invariant for (nondiscrete) matroids. Finally I will discuss T2 and show that several multigraded components are trivial. This talk is based on a joint project with William Bitsch. 02.05.2022 Lucie Devey (Goethe Universität, Frankfurt am Main) Titel: The polytopes are not what they seem Abstract: We know a lot on toric divisors D on toric varieties. In particular, their data is equivalent to the data of a polytope. And geometry comes on stage. With this polytope we may know instantaneously a basis of the global sections of O(D) and a lot of positivity properties: we may know if D is globally generated, ample, nef, movable, big, ... In view of the success of Newton polytopes, we may want to generalise these objects to toric vector bundle of any rank. That is the aim of parliaments of polytopes. It is a collection of polytopes (as before) indexed by elements of a representable matroid. And combinatorics comes on stage. In this talk, I will present you the construction of the parliament of polytopes of a toric vector bundle and give an overview of the astounding wealth of information it contains. 25.04.2022 Martí Salat Moltó (Universitat de Barcelona) Titel: Multigraded regularity of equivariant reflexive sheaves on a toric variety Abstract: Using the theory of Klyachko filtrations for equivariant reflexive sheaves on a toric variety, we construct a collection of lattice polytopes which codifies its global sections. By means of this approach the description of the multigraded Hilbert polynomial of any equivariant reflexive sheaf is translated to a lattice-point counting problem. In the case of smooth projective varieties of Picard number 2, this problem can be effectively solved. On one hand this yields a bound for the multigraded regularity index, and on the other hand a method to compute the multigraded Hilbert polynomial effectively. Winter Semester 2021/2022 14.02.2022 Talk starts at 5:15pm ONLY via WebEx! Jamie Pommersheim (Reed College, Portland) Titel: Triangulations of a Square and Monsky Polynomials Abstract: In 1970, Paul Monsky proved that a square cannot be dissected into an odd number of triangles of equal area. This theorem raises the more general question of what other restrictions there might be on the areas of the triangles in a dissection of a square. It turns out that if one fixes the combinatorics of the dissection, then there is a single irreducible polynomial that must be satisfied by the areas of the triangles in the dissection. These Monsky polynomials and mysteries surrounding them will be the main focus of this talk. 07.02.2022 Wendelin Lutz (Imperial College London) Titel: A geometric proof of the classification of T-polygons Abstract: One formulation of mirror symmetry predicts (omitting a few adjectives) a one-to-one correspondence between equivalence classes of lattice polygons and deformation families of del Pezzo surfaces. Lattice polygons that correspond to smooth Del Pezzo surfaces are called T-polygons and have been classified by Kasprzyk-Nill-Prince using combinatorial methods, thereby verifying the conjecture in the smooth case. I will give a new geometric proof of their classification result. 17.01.2022 Severin Barmeier (Universität Köln) Titel: A combinatorial approach to associative deformations and applications to geometry Abstract: I will present a combinatorial approach to the deformation theory of finitely generated associative algebras, or more generally of path algebras of finite quivers with relations. The main idea behind this approach is to replace the bar resolution, which gives rise to the Hochschild cochain complex in the classical deformation theory developed by Gerstenhaber, by a smaller combinatorial resolution and use the resulting cochain complex to describe the associative deformations. Doing this, one obtains a surprisingly workable description of the deformation theory, which can be used to compute concrete examples, and also allows one to give criteria for passing from formal to "actual" deformations, where the formal deformation parameters are evaluated to constants. I will give an overview of various applications in algebraic geometry, where associative deformations are closely related to the quantization of Poisson structures. In particular, one can use this combinatorial approach to give an explicit description of the deformation theory of categories of coherent sheaves, which can be related to noncommutative varieties in noncommutative projective geometry, and can also be used as a tool for obtaining noncommutative deformations of singularities. This talk will largely based on arXiv:2002.10001 and arXiv:2107.07490 both joint with Zhengfang Wang. 10.01.2022 Only via webEx! Frederik Witt (Universität Stuttgart) Titel: The structure of exceptional sequences on toric varieties of Picard rank two Abstract: For a smooth projective toric variety of Picard rank two we classify all exceptional sequences of invertible sheaves which have maximal length. In particular, we prove that unlike non-maximal sequences, they (a) remain exceptional under lexicographical reordering (b) satisfy strong height constraints in the Picard lattice (c) are full, that is, they generate the derived category of the variety. 13.12.2021 No talk! 06.12.2021 PLEASE NOTE THAT THIS TALK WILL BE ONLINE ONLY! Hendrik Süß (Friedrich-Schiller-Universität Jena) Titel: The normalised volume of singularities and the Mahler volume of polytopes Abstract: In my talk I will discuss the notion of normalised volume for log terminal singularities. For the special case of toric singularities this turns out to be closely related to the notions of Mahler volume and Santaló point in convex geometry. I will explain how well-known facts from convex geometry can be utilised to deduce non-trivial statements about toric singularities. The Mahler conjecture will have a short appearance as well. 29.11.2021 No talk! 22.11.2021 Luis Sola (University of Trento) Titel: Do rational homogeneous spaces dream of birational geometry? Abstract: In this talk I will present some results obtained in the framework of a collaboration with G. Occhetta, E. Romano, and J. Wisniewski. We will discuss some properties of smooth complex projective varieties admitting torus actions, and their relation with certain birational transformations. I will also present some examples constructed upon rational homogeneous spaces. 15.11.2021 Milena Wrobel (University of Oldenburg) Titel: Smooth Fano varieties with torus action of complexity two Abstract: We study smooth Fano varieties with torus action by using their description via a specific rational quotient, the so called maximal orbit quotient (MOQ). In the case of torus actions of complexity one, the MOQ turns out to be a projective line, having points as its critical configuration. In this talk we focus on torus actions of complexity two, which turns the MOQ into a surface allowing a bunch of possible quotients. We shortly recall the case where the MOQ is a projective plane with a hyperplane arrangement as its critical configuration. Then, going one step further, we replace the projective plane with$\PP^1 \times \PP^1$and give first classification results in this setting. 08.11.2021 Leonid Monin (Max Planck Institute for Mathematics in the Sciences, Leipzig) Titel: Gorenstein algebras and computations of cohomology rings Abstract: It was observed by Pukhlikov and Khovanskii that the BKK theorem implies that the volume polynomial on the space of polytopes is the Macaulay generator of the cohomology ring of a smooth projective toric variety. This provides a way to express the cohomology ring of a toric variety as a quotient of the ring of differential operators with constant coefficients by the annihilator of an explicit polynomial. The crucial ingredient of this observation is an explicit expression for the Macaulay generator of graded Gorenstein algebras generated in degree 1. In my talk I will explain recent results on an explicit expression for the Macaulay generator of an arbitrary algebra with Gorenstein duality and how to use them for the computation of cohomology rings. I will illustrate this approach with a computation of cohomology rings of toric bundles. If time permits I will also explain the consequent computation of the ring of conditions for horospherical homogeneous spaces. 01.11.2021 Jakub Koncki (University of Warsaw) Titel: Stable basis and boundness of the motivic Chern class Abstract: The motivic Chern class is a generalization of the Chern class of tangent bundle to singular setting. I will present this class and prove that it is in some sense bounded. This result allows to compare the motivic Chern class with the stable basis. 25.10.2021 Alex Kuronya (Uni Frankfurt/Main) (WebEx talk at 17:15h!) Titel: Functions on Newton-Okounkov bodies Abstract: The purpose of this talk is to give a brief overview of the construction of concave transforms of filtrations on graded rings, which results in a class of interesting concave functions on Newton-Okounkov bodies. In addition to the fundamental properties of concave transforms, we discuss examples, structure results, and future research directions. Summer Semester 2021 The talks will be held on Mondays at 16:15 via Webex (https://www.webex.com). Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m557416b71c55a1d50a22cc33627f21b3 Meeting number: 121 099 5804 Password: Toric*****Variety without stars. The schedule of the talks, with abstracts, appears at https://researchseminars.org/seminar/algebraic_geometry_FU The schedule of the talks, without abstracts, is below. 12.04.2021 Simon Felten (Universität Mainz) The logarithmic Bogomolov-Tian-Todorov theorem 19.04.2021 Mattia Talpo (University of Pisa) 26.04.2021 Milena Wrobel (University of Oldenburg) 03.05.2021 Jonathan Lai (Imperial College London) 10.05.2021 No talk - you can watch a talk at the online seminar "Algebra, Geometry, and Combinatorics" https://sites.google.com/view/agconline/ or a talk at the conference "Groupes et géométrie algébriques" http://webusers.imj-prg.fr/~frederic.han/lg2021/ 17.05.2021 Lara Bossinger (UNAM Oaxaca) 24.05.2021 No talk - Pfingstmontag 31.05.2021 Marian Aprodu (University of Bucharest) 07.06.2021 Enrica Mazzon (MPI Bonn) 14.06.2021 Aniket Shah (Ohio State University) 21.06.2021 Matteo Varbaro (University of Genova) 28.06.2021 Thomas Krämer (Humboldt-Universität zu Berlin) 05.07.2021 Rita Pardini (University of Pisa) 12.07.2021 Winter Semester 2020/2021 Schedule: 02.11.2020 kein Seminar 09.11.2020 Karin Schaller (FU-Berlin) Stringy Chern classes 16.11.2020 Amelie Flatt (HU-Berlin) Displaying Ext 23.11.2020 Noah Giessing (Potsdam) Localization in algebraic K-theory 30.11.2020 Noah Giessing (Potsdam) Rational homogeneous spaces 07.12.2020 TALK POSTPONED! Alvaro Liendo (Universidad de Talca, Chile) 14.12.2020 Lena Walter (FU-Berlin) Toric Newton-Okounkov functions We initiate a combinatorial study of Newton-Okounkov functions on toric varieties with an eye on the rationality of asymptotic invariants of line bundles. In the course of our efforts we identify a combinatorial condition which ensures a controlled behavior of the appropriate Newton-Okounkov function on a toric surface. Our approach yields the rationality of many Seshadri constants that have not been settled before. This is joint work with Christian Haase and Alex Kueronya. 04.01.2021 Livia Campo (Nottingham) Sarkisov links for index 1 Fano 3-folds in codimension 4 As possible outcomes of the Minimal Model Program, Mori fibre spaces play a crucial role in Birational Geometry, and is therefore important to study how they connect to each other. The Sarkisov Program provides tools to analyse birational transformations between Mori fibre spaces. In this talk I will examine the case of codimension 4 Fano 3-folds with Fano index 1 and explain how machineries likegraded rings and GIT can help to study their birational classes. This study also leads to some conclusions regarding their Picard rank. webEx; Meeting number: 121 255 9136, Meeting password: JcgRQneN525 Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m27d010244c4481dca71d7e5fe9d4ac41 11. 01.2021 Joaquin Moraga (Princeton) The Jordan property for local fundamental groups For a finite subgroup G of GL_n(C), we can find a normal abelian subgroup A<G so that its index in G is bounded by a constant c(n), which only depends on n. In this talk, we prove that a similar statement holds for the local fundamental group of n-dimensional klt singularities. Then, we show applications of this statement to the study of iteration of Cox rings of Fano type varieties. webEx; Meeting number: 121 255 9136, Meeting password: JcgRQneN525) Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m27d010244c4481dca71d7e5fe9d4ac41 18.01.2021 PLEASE NOTE THAT THE DATE OF THIS TALK IS NOW 15.02.2021: Vladimir Lazić (Saarbruecken) Which properties of the canonical class depend only on its first Chern class 25.01.2021 Due to illness this talk will take place on March 1st. Valentina Kiritchenko (Moscow) Push-pull operators and Newton-Okounkov polytopes 01.02.2021 Hendrik Suess (Manchester) TBA webEx; Meeting number: 121 255 9136, Meeting password: JcgRQneN525) Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m27d010244c4481dca71d7e5fe9d4ac41 08.02.2021 Alvaro Liendo (Talca, Chile) Characterization of algebraic varieties by their groups of symmetries webEx; Meeting number: 121 255 9136, Meeting password: JcgRQneN525) Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m27d010244c4481dca71d7e5fe9d4ac41 15.02.2021 Vladimir Lazić (Saarbruecken) Which properties of the canonical class depend only on its first Chern class? Given a projective variety X with mild singularities and a line bundle L on it, it is a natural question to determine which properties of L are encoded by its first Chern class. I will argue that most of the interesting properties of the canonical bundle of X, such as its effectivity or semiampleness, are indeed almost always encoded by its first Chern class. The results are a consequence of the Minimal Model Program and Hodge theory, and are new even on surfaces. This is joint work with Thomas Peternell. webEx; Meeting number: 121 255 9136, Meeting password: JcgRQneN525) Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m27d010244c4481dca71d7e5fe9d4ac41 22.02.2021 Michel Brion (Grenoble) Homomorphisms of algebraic groups: representability and rigidity The talk will discuss the following questions. Given two algebraic groups G and H (over the complex numbers simplicity), is there a "moduli space" M for the homomorphisms from G to H? In the affirmative, how to describe the H-orbits on M, where H acts by conjugation of homomorphisms? By a result of Grothendieck, M exists when G is diagonalizable and H is linear; then every H-orbit in M is open. On the other hand, M does not exist when G is the additive group and H is the multiplicative group. We will sketch how to extend Grothendieck's results to the case where G is reductive (and H arbitrary). webEx; Meeting number: 121 255 9136, Meeting password: JcgRQneN525) Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m27d010244c4481dca71d7e5fe9d4ac41 01.03.2021 Valentina Kiritchenko (Moscow) Push-pull operators and Newton-Okounkov polytopes A classical result of Schubert calculus is an inductive description of Schubert cycles using divided difference (or push-pull) operators in Chow rings of projective bundles. We define convex geometric analogs of push-pull operators by replacing Chow rings with polytope rings (Khovanskii-Pukhlikov rings). We apply push-pull operators to describe Newton-Okounkov polytopes of Schubert varieties and of Bott-Samelson varieties. In particular, Grossberg-Karshon cubes and Minkowski sums of Fegin-Fourier-Littelmann-Vinberg polytopes can be constructed inductively using convex grometric push-pull operators. All definitions will be given in the talk. New Link! https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=mc13de3b6c3fa3fb9d708d2e6e8bc2d25https Meeting number: 121 021 0842 Password: APvpnWM4N82 Sommer Semester 2020 Schedule 20.04.2020 Johannes Hofscheier (Nottingham) (Webex talk--details below) A geometric proof of the generalised Mukai conjecture for horospherical Fano varieties Abstract: Horospherical varieties naturally generalise toric and flag varieties. They form a rich class of algebraic varieties admitting an action by a reductive group with an open dense orbit. In this talk, I will present joint work in progress with Giuliano Gagliardi on a geometric proof of the generalised Mukai conjecture for horospherical Fanovarieties. Our approach uses a geometric characterisation of topic varieties using log pairs which was conjectured by Shokurov and recently proven byBrown-McKernan-Svaldi-Zong. Meeting-Link: https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m7e8fe38c0f6e7005a499ba5f085feee5 (Meeting number 849 418 444, password wTbiMZCb898). 27.04.2020 Elena Martinengo (Turin) (Webex-talk--details below) On degeneracy loci of equivariant bi-vector fields on a smooth toric variety. On a smooth toric variety$X$of dimension$n$, we study equivariant bi-vector fields, i.e. global sections of the second symmetric power of the homolorphic tangent bundle that are equivariant with respect to the toric action. We are interested in the degeneracy loci, that are the loci in which therank of such a bi-vector field is less or equal some integer$k$. In particular, in the spirit of a Bondal conjecture, we prove that the locus where the rank of an equivariant bi-vector field is$\leq 2k$is not empty and has at least a component of dimension$\geq 2k+1$, for all integers$k > 0$such that$2k < n$. The same is true also for$k = 0\$, if the toric variety is smooth and compact. While for the non compact case, the locus in question has to be assumed to be non empty.

https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m7e8fe38c0f6e7005a499ba5f085feee5 (Meeting number 849 418 444, password wTbiMZCb898)

18.05.2020

Rostislav Devyatov (Bonn) (Webex talk--details below)

https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m7e8fe38c0f6e7005a499ba5f085feee5 (Meeting number 849 418 444, password wTbiMZCb898)

25.05.2020

Andreas Hochenegger (Pisa)  (Webex talk--details below)

Ulrich bundles on projective bundles

Abstract: Given some smooth projective variety X, it is a difficult question whether X admits an Ulrich bundle, which are locally free sheaves fulfilling strong conditions on its cohomology. In this talk I will present some constructions of Ulrich bundles in the case that X itself is a projective bundle. This is about a joint work in progress with Klaus Altmann.

https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m7e8fe38c0f6e7005a499ba5f085feee5 (Meeting number 849 418 444, password wTbiMZCb898)

01.06.2020

No talk! Pentecost holiday.

08.06.2020

Lutz Hille (Muenster)  (Webex talk--details below)

Inverse limits of moduli of quiver representations, Chow quotients and Losev-Manin moduli spaces

Abstract: Motivated by the construction of the compactifications of the moduli space of marked curves of genus zero with n marks (often called Mumford-Knudsen moduli space) we consider a similar construction for certain quiver representations. This construction uses King's construction of the moduli space of \Theta-stable quiverrepresentations  and a certain inverse limit over all possible weights \Theta.

In a special case we recover the Mumford-Knudsen, the Losev-Manin and all the Hassett compactifications for a certain choice of a quiver, a dimension vector and a certain set of Weights.

In this talk I consider the toric situation, that started as joint work with Altmann about 25 years ago and is still a challenge from a combinatorial point of view. In a particular situation we get the Losev-Manin compactification, however, using quivers, it is obvious how to generalize this approach (joint work with Mark Blume).

https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m7e8fe38c0f6e7005a499ba5f085feee5 (Meeting number 849 418 444, password wTbiMZCb898)

15.06.2020

Jan Stovicek (Prague) (Webex talk--details below)

Pure-injective quasi-coherent sheaves on elliptic curves

Abstract: The aim of the talk is to explain a joint contribution with Alessandro Rapato to the classification of indecomposable pure-injective quasi-coherent sheaves over the elliptic curves. This work is contained in his thesis (https://http://eprintsphd.biblio.unitn.it/3769/1/PhD_Thesis_Rapa.pdf)

The notion of pure-injective (also known under the name "algebraically compact") module originated as a key concept in model theory of modules and extends to categories sheaves. Coherent sheaves on smooth projective schemes over a field are pure-injective, as are finite dimensional modules over finite-dimensional algebras, but the conversei s usually not true. In some sense, pure-injective quasi coherent sheaves should be the next best understandable sheaves to coherent ones, analogously to the case of modules.

The situation is completely understood for the projective line, and here we focus on elliptic curves. Building on the work of Reiten and Ringel (who introduced the notion of slope for any indecomposable quasi-coherent sheaf) and in the context of a recent work of Kussin and Laking (arXiv:1911.02485), we found large families of indecomposable pure-injective quasi-coherent sheaves of irrational slope.

https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m7e8fe38c0f6e7005a499ba5f085feee5 (Meeting number 849 418 444, password wTbiMZCb898)

22.06.2020

Thomas Eckl (Liverpool) (Webex talk--details below)

Nagata's Conjecture, Kaehler packings and toric degenerations

Abstract: Interpolation problems in the (complex projective) plane are a classical topic still posing many wide open conjectures, like Nagata's Conjecture. In modern terminology, these conjectures can be interpreted as predictions on local positivity on polarized algebraic varieties. In this talk, we first motivate that measuring local positivity is closely connected to solving packing problems in symplectic and Kaehler geometry. In particular, we show that Nagata's Conjecture is equivalent to a prediction on the maximal possible size of multiple balls packed into the complex projective plane respecting the Kaehler structure. Finally, we indicate how toric degenerations can be used to construct Kaehler packings explicitly.

https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m7e8fe38c0f6e7005a499ba5f085feee5 (Meeting number 849 418 444, password wTbiMZCb898)

29.06.2020

Hal Schenck (Auburn) (Webex talk--details below)

Milnor ring of a singular projective hypersurface V(F)

Abstract: For a reduced hypersurface V(f) in P^n of degree d, the Milnor algebra M(f) is a quotient of the polynomial ring by the Jacobian ideal of f. The Castelnuovo-Mumford regularity of the Milnor algebra M(f) is well understood when V(f) is smooth, as well as when V(f) has isolated singularities. We study the regularity of M(f) when V(f) has a positive dimensional singular locus. In certain situations, we prove that the regularity is bounded by (d?2)(n+1), which is the degree of the Hessian polynomial of f. However, this is not always the case, and we prove that the regularity can grow quadratically in d. I will also describe several famous theorems about why one cares about M(f), including results from hyperplane arrangement cohomology and Hodge theory

https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m7e8fe38c0f6e7005a499ba5f085feee5 (Meeting number 849 418 444, password wTbiMZCb898)

13.07.2020

Nathan Ilten (SFU, Burnaby, Canada) (Webex talk--details below)

Fano schemes for complete intersections in toric varieties

Abstract: The study of the set of lines contained in a fixed hypersurface is classical:Cayley and Salmon showed in 1849 that a  smooth cubic surface contains 27 lines, and Schubert showed in 1879 that a generic quantic threefold contains 2875 lines. More generally, the set of k-dimensional linear spaces contained in a fixed projective variety X itself is called the k-th Fano scheme of X. These Fano schemes have been studied extensively when X is a general hypersurface or complete intersection in projective space. In this talk, I will report on work with Tyler Kelly in which we study Fano schemes for hypersurfaces and complete intersections in projective toric varieties. In particular, I'll give criteria for the Fano schemes of generic complete intersections in a projective toric variety to be non-empty and of "expected dimension". Combined with some intersection theory, this can be used for enumerative problems, for example, to show that a general degree (3,3) hypersurface in the Segre embedding of P2?P2P2?P2 contains exactly 378 lines.

https://sfu.zoom.us/j/95686986289?pwd=WUVKekRQV0duU2Y1V0NwSTFYQ0xPUT09
Meeting ID: 956 8698 6289

Winter Semester 2019/2020

Unless otherwise specified talks take place in room 119 (Arnimallee 3) at 16:15

Schedule

17.10.2019

Luis Sola Conde (Trento)

Characterizing rational homogeneous varieties.

Abstract: I
n this talk we will discuss possible characterizations of homogeneity within the framework of Fano varieties. More concretely, we will first examine the relation among the homogeneity of a Fano variety, and the positivity of its tangent bundle. In the second part of the talk, we will consider fiber bundles on rational homogeneous varieties, and study how uniformity could imply homogeneity in this setting.

24.10.2019

Andrzej Weber (Warschau)

Elliptic characteristic classes of Schubert varieties and Hecke-type algebra

We consider the homogeneous spaces G/P, where G is a complex reductive group, and P its a parabolic subgroup. We study characteristic classes of Schubert varieties in cohomology or K theory. The elliptic classes defined by Borisov and Libgober are of special interest. To compute them we need to understand the canonical divisors of Schubert varieties. This can be obtained by analysis of Bott-Samelson resolutions. As the output we obtain inductive formulas extending the classical results of Demazure, Beilinson-Gelfan-Gelfand, Lusztig and Lascoux-Schutzenberger.

This is a joint work with Shrawan Kumar and Richard Rimanyi.

31.10.2019

Eleonora Romano (Warschau)

Torus actions on projective manifolds: Combinatorics vs Birational Geometry

Abstract: In this talk we focus on one-dimensional torus actions on complex smooth projective varieties of arbitrary dimension. On one hand, we introduce some combinatorics tools which we need in our situation, on the other hand we discuss how Birational Geometry takes his role. In particular, we review the classical adjunction theory from a new perspective given by the torus acting. Moreover we see how this new approach allows to get some classification results for special varieties called of "small bandwidth". As an application of our results, we will conclude by putting them in the context of the classification of contact Fano varieties of high dimension. This talk is a joint work with J. Wisniewski, and includes a work in progress with G. Occhetta, L. Sola' Conde and J. Wisniewski.

14.11.2019

Simon Telen (KU Leuven, Belgium/TU-Berlin)

Numerical Root Finding via Cox Rings

Abstract: In this talk, we consider the problem of solving a system of (sparse) Laurent polynomial equations defining finitely many nonsingular points in a compact toric variety. The Cox ring of this toric variety is a generalization of the homogeneous coordinate ring of projective space. We work with multiplication maps in graded pieces of this ring to generalize the eigenvalue, eigenvector theorem for root finding in affine space. We present a numerical linear algebra algorithm for computing the corresponding matrices, and from these matrices a set of homogeneous coordinates of the roots of the system. Several numerical experiments show the effectiveness of the resulting method, especially for solving (nearly) degenerate, high degree systems in small numbers of variables.

21.11.2019

Ilya Smirnov (Stockholm)

Lech's inequality and its combinatorics.

Abstract: Multiplicity and colength are two most basic invariants of an m-primary ideal of a local ring. These two invariants can differ drastically, but in 1960 Lech found that the ratio multiplicity(I)/colength(I) is bounded uniformly by the multiplicity of the ring.

If dimension is not two, this inequality is never sharp. In a joint work with Craig Huneke and Javid Validashti, we conjectured an improvement and made a certain progress.

In my talk, I will discuss these results and will give all necessary background details. I will focus on an existing connection with enumerative combinatorics and will give a combinatorial proof of Lech's inequality and explain a combinatorial form of our conjecture.

Wednesday!!! 04.12.2019

Olivier Benoist (Paris)

Rational curves in real rationally connected varieties.

Abstract: Can one approximate a loop drawn in the real locus of a real rationally connected variety by the real locus of a rational curve lying on it? We will give a positive answer in particular cases, including cubic hypersurfaces, intersections of two quadrics, and compactifications of homogeneous spaces. This is joint work with Olivier Wittenberg.

23.01.2020

Alessio Corti (London)

Smoothing Gorenstein toric affine 3-folds: some Conjectures

Abstract. I present a conjectural description of the smoothing components of the deformation space. I discuss context, evidence, and relation to mirror symmetry. Work with Matej Filip and Andrea Petracci.

30.01.2020

No talk: NoGaGS in Hannover

06.02.2020

Irem Portakal (Magdeburg)

Combinatorial aspects of Kazhdan-Lusztig varieties

Abstract: In this talk, we present a combinatorial introduction to Kazhdan-Lusztig (KL) varieties in terms of Rothe diagrams. This enables us to understand the so-called usual torus action on them in terms of simple directed graphs. Moreover there is a one-to-one correspondence between KL varieties and the affine neighborhoods of torus fixed points in the Schubert variety in the full flag variety. We utilize existing results about the usual torus action on the Schubert variety in order to determine the complexity of the torus action on KL varieties. This is joint work with Donten-Bury and Escobar.

13.02.2020

Roser Homs Pons (Leipzig)

Computing minimal Gorenstein covers

Abstract: We analyze and present an effective solution to the minimal Gorenstein cover problem: given a local Artin k-algebra A =k[[x_1,...,x_n]]/I, compute an Artin Gorenstein k-algebra G = > k[[x_1,...,x_n]]/J such that l(G)−l(A) is minimal. We approach the problem by using Macaulay’s inverse systems and a modification of the integration method for inverse systems to compute Gorenstein covers. We propose new characterizations of the minimal Gorenstein cover and present a new algorithm for the effective computation of the variety of all minimal Gorenstein covers of A for low Gorenstein colength.

Summer Semester 2019

Unless otherwise specified talks take place in room 119 (Arnimallee 3) at 16:15

Schedule

!! Tuesday, 23.04.2019  at 12: 15 in room 210 (Arnimallee 3) !!

Duco van Straten (Mainz)

Continuous algebraic geometry: exploring fractional dimension.

Abstract: The concept of interpolation to a continuous variable is very old, and has turned out to be very useful, the Gamma-function being an example in case. In this talk I will report on joint work with V. Golyshev on projective spaces and Grassmanians of fractional dimension and the algebraic geometry related to it.

02.05.2019

Jarek Wisniewski (Warsaw)

Combinatorics of C* action and varieties of small bandwidth.

Abstract: Motivated by LeBrun-Salamon conjecture about quaternion-Kahler manifolds we study polarized pairs (X,L) with an action of C* on the complex manifold X such that its general orbit has small degree with respect to the ample line bundle L. For this purpose we describe adjunction morphism for (X,L) using combinatorics arising from the action of C*. In passing we get a reinterpretation of classical objects in projective and birational geometry like Severi varieties and special Cremona tranformations. I will report on a joint work with Romano and Ochetta, Sola Conde.

16.05.2019

Luca Battistella (MPI, Bonn)

Reduced Gromov-Witten invariants and singularities of genus one

Abstract: Classical enumerative geometry produced beautiful theorems such as: every smooth cubic surface contains exactly 27 lines. The subject has been reinvigorated since the introduction of the moduli space of stable maps, which, for example, allowed Kontsevich to solve a long-standing problem: the number of rational plane curves of degree d passing through 3d-1 general points. More generally, it made it possible to answer many questions on rational curves in projective complete intersections.

Dealing with curves of higher genus is more difficult, because the moduli spaces are not so well behaved. I will explain this with an example in genus one. We will then see two different approaches to the problem: a desingularisation of the moduli space, which led to the definition of reduced invariants, due to Li-Vakil-Zinger, and alternative compactifications obtained by Viscardi by allowing the source curve to acquire a singularity of genus one. For the quintic threefold, the two approaches are put in relation in joint work with Carocci and Manolache.

If time permits, I will discuss how log geometry - a far-fetching generalisation of toric geometry which is often useful to single out and improve the main component of moduli spaces - enters the picture, both by producing natural contractions to singular curves, and allowing us to study the relative problem, i.e. a count of curves tangent to a hyperplane section. This is joint work with Nabijou and Ranganathan, based on a ground-breaking paper of Ranganathan, Santos-Parker, and Wise.

23.05.2019

Matej Filip (Mainz)

The versal deformation for toric varieties in special lattice degrees.

Abstract: I will describe the versal deformation for non-solated Gorenstein toric varieties in the Gorenstein lattice degree.

06.06.2019

Klaus Altmann (FU-Berlin)

Deformation of toric singularities by universal extensions of semigroups

Abstract: We show how extensions of semigroups lead to deformations of toric singularities. On the level of semigroups, universal extensions exist, and they lead to versal deformations of the toric singularities in a prescribed multidegree. This is joint work with Alexandru Constantinescu (FUB) and Matej Filip (Mainz).

20.06.2019

Christian Sevenheck (Chemnitz)

Hodge ideals for certain free divisors

Abstract: In recent years, there has been renewed interest in the theory of mixed Hodge modules, mainly motivated by questions from birational geometry. In particular, Mustata and Popa have defined the so-called Hodge ideals, which describes to a certain extend to Hodge filtration on the complement of a singular divisor. In this talk, I will explain what Hodge ideals are and then discuss a particular case, namely that of certain free divisors, where quite explicit statements about the Hodge filtration can be made. This is joint work with Luis Narvaez Macarro (Sevilla) and Alberto Castaño Domínguez (Santiago de Compostella).

Rekha Thomas (Seattle)

Toric slack ideals and psd-minimal polytopes

Abstract: Slack ideals of polytopes are saturated determinantal ideals that encode all realizations of polytopes in the same combinatorial class. The simplest of these ideals are toric and it is an open question as to which polytopes give rise to toric slack ideals. However, we do know many special properties of polytopes with toric slack ideals and in this talk I will explain these results. They are related to both projectively unique polytopes and polytopes that admit the smallest  possible lifts into psd cones, two rather unrelated concepts it seems. There are several open questions in this circle of ideas which I will mention.

Archiv (WS 05/06 - WS 09/10)

Archiv (SoSe 2011...)