Oberseminar Algebraische Geometrie
We meet regularly on Wednesdays in the building E2 4, room SR 10, starting at 2pm.
NEWS (Wintersemester 2021/22)
We meet regularly on Wednesdays in SR10 or online in Zoom, starting at 2pm.
Vladimir Lazić (UdS): The Nonvanishing problem for varieties with nef first Chern class
Sokratis Zikas (Basel): Rigid birational involution of projective space and cubic 3-folds
|04.05.2022||Quentin Posva (EPFL): Sources and springs in positive characteristic, and applications|
The singular varieties appearing in the KSBA moduli theory of canonically polarised manifolds are called stable varieties: these are the varieties with an ample canonical sheaf, and semi-log canonical singularities. In particular, stable varieties need not be normal. Various technical challenges arise: among them is the failure of the MMP for non-normal varieties. In characteristic zero, a solution to this problem was developed by Kollár, who established a dictionary between stable varieties and their normalisations. This dictionary relies on the theory of sources and springs of log canonical centers. In this talk I will recall this theory, and explain how it can be generalised to positive characteristic in low dimensions. Then I will discuss some applications and open questions.
|11.05.2022||Frank-Olaf Schreyer (UdS): Extensions of paracanonical curves of genus 6|
Two-regular subschemes of Pn have been studied and classified by Bertini, Eisenbud, Harris et. al. for a long time. The study of 3-regular subschemes whose ideal is generated by cubics was initiated by Sijong Kwak and co-authors. The first open case for their classification are 3-regular varieties of codimension 3 and degree 10. In case of curves these are paracanonical embedded curves.
A non-degenerate variety Y⊆Pn+1 of dimension dim X+1, which is not a cone, is an extension of a variety X⊆Pn if there exists an hyperplane H⊆Pn+1 such that X≅Y∩H.
In this talk I will report on joint work with Hoang Truong where we classify extensions of a general paracanonical curve of genus 6 and degree 10 in P4. Our main result says that there are 26 different families of extensions for a given general paracanonical curve C⊆P4.
Jingjun Han (Fudan): Birational boundedness of rationally connected log Calabi-Yau pairs with fixed index
|08.06.2022||Fulvio Gesmundo: TBA|
|29.06.2022||Lukas Braun (Freiburg): TBA|
Nikolaos Tsakanikas (UdS): Termination of flips for pseudo-effective 4-folds
Zhixin Xie (UdS): Rationally connected threefolds with nef anticanonical divisor
|10.11.2021||Giacomo Mezzedimi (Hannover): The Potential Hilbert Property on Enriques and K3 surfaces|
The problem of determining "how many" rational points an algebraic variety has lies at the heart of Number Theory and Arithmetic Geometry. One way to formalize this idea is provided by the (weak) Hilbert Property, a modern generalization of Hilbert's classical irreducibility theorem, concerning the distribution of the rational points on the variety and on its finite covers. An important conjecture by Campana and Corvaja-Zannier predicts a strong interplay between the Hilbert Property and the density of rational points on the variety. In this talk I will report on a joint work with D. Gvirtz-Chen in which we solve this conjecture for Enriques surfaces and for a wide class of K3 surfaces.
|17.11.2021||Stefano Filipazzi (EPFL): On the boundedness of elliptic Calabi-Yau threefolds|
In this talk, we will discuss the boundedness of Calabi-Yau threefold admitting an elliptic fibration. First, we will survey some ideas to address weak forms of boundedness of varieties admitting an elliptic fibration. Then, we will switch focus to Calabi-Yau varieties and discuss how the Kawamata-Morrison cone conjecture comes in the picture when studying boundedness properties for this class of varieties. To conclude, we will see how this circle of ideas applies to the case of elliptic Calabi-Yau threefolds. This talk is based on work joint with C.D. Hacon and R. Svaldi.
|24.11.2021||Giovanni Mongardi (Bologna): Irrational Gushel Mukai threefolds|
We construct an explicit complex smooth Fano threefold with Picard number 1, index 1 and degree 10 (a Gushel-Mukai threefold) and prove that it is not rational by showing that its intermediate jacobian has a faithful PSL(2,F11) action. The construction is based on a very special double EPW sextic. This is joint work with O. Debarre.
Christian Lehn (TU Chemnitz): Singular varieties with trivial canonical class
|Shigeyuki Kondo (Nagoya): Coble surfaces with finite automorphism group|
A Coble surface is a rational surface which appears as a degeneration of Enriques surfaces. Thus Coble surfaces inherit many properties of Enriques surfaces and complement the theory of Enriques surfaces. A general Coble surface has an infinite group of automorphisms and hence it is natural to classify Coble surfaces with finite automorphism group as in the case of Enriques surfaces. We will give the classification of such surfaces: there are exactly 9 isomorphism classes of such surfaces in characteristic p≠2 and 9 families in p=2. In case of p=2, this is a joint work with Toshiyuki Katsura.
Isabel Stenger (UdS): On the numerical dimension of Calabi-Yau threefolds of Picard number 2
Yvonne Neuy (UdS): Factorization of polynomials
Hans-Christian Graf von Bothmer (Hamburg): Rigid, not infinitesimally rigid surfaces with ample canonical bundle
|14.07.2021||Firoozeh Aga (UdS/Fraunhofer ITWM): Algorithmic generation of automorphism groups of K3-surfaces|
A complex K3 surface S is a compact connected Kähler complex surface, with trivial canonical bundle and H1(S,OS)=0. The Fermat quartic, given by the equation V(x04+ x14+x24+x34)∊P3, is an example of such a surface. In this talk, we begin by introducing some important properties of K3 surfaces. Next, we will look at the notions from Lattice theory, and other invariants required to compute the automorphism groups of these surfaces. We will briefly discuss the algorithm by R. E. Borcherds to compute these automorphism groups. Following which we will look at a generalization of his algorithm by Ichiro Shimada. I will also give a list of examples for which the automorphism groups have already been computed using these methods. Next, I will present the computational tools (GPI-Space, Julia, and their embedding) required to parallelize algorithms and explore different components of GPI-Space. Using a small example, we will design a Petri net and see how we can benefit (in terms of reduced computation time) from a ’well designed’ Petri net. Finally, I will present a list of tasks that I am currently working on. These could be potentially used to improve the algorithms mentioned above by using different parallel programming models.
|Keiji Oguiso (Tokyo): Real forms of a smooth complex projective variety|
This talk is based on a series of joint works with Professors Tien-Cuong Dinh and Xun Yu. After Lesieutre, there were several works related to a long standing open question since Kharlamov: Is there a smooth complex projective rational surface with infinitely many real forms up to isomorphisms, i.e., with infinitely many ways to describe it by a system of equations with real coefficients up to isomorphism over the real number field? After a brief introduction to the real form problem, and in particular, a way to reduce the problem to a problem on the conjugacy classes of involutions in the automorphism group, I would like explain our affirmative answer to the question, with a proof using a special Kummer surface of product type and its rich geometry. We then explain the results in higher dimensions and/or non-negative Kodaira dimensions, together with related open problems.
|28.07.2021||Yvonne Neuy (UdS): Factorization of polynomials, II|
|04.11.2020||Yeongrak Kim (UdS): Ulrich bundles on cubic fourfolds|
|11.11.2020||Andreas Braun (Durham): String Compactifications and Arithmetic |
String theory gives a strong motivation for studying elliptic fibrations and Hodge cycles on K3 surfaces, as well as higher-dimensional Calabi-Yau varieties. After explaining the relevant data of the setup and giving a rudimentary physics dictionary, I will review some recent results and highlight open problems.
|02.12.2020||Ann Sophie Cenkel (UdS): Borsuk-Ulam Theorem and its applications|
|09.12.2020||Edgar Costa (MIT): Effective obstruction to lifting Tate classes from positive characteristic |
We present an algorithm that takes a smooth hypersurface over a number field and computes a p-adic approximation of the obstruction map on the Tate classes of a finite reduction. This gives an upper bound on the "middle Picard number" of the hypersurface. The improvement over existing methods is that our method relies only on a single prime reduction and gives the possibility of cutting down on the dimension of Tate classes by two or more. The obstruction map comes from p-adic variational Hodge conjecture and we rely on the recent advancement by Bloch-Esnault-Kerz to interpret our bounds. This is joing work with Emre Sertoz.
|Klaus Altmann (FU Berlin): Displaying the universal extension of toric line bundles |
Line bundles L on projective toric varieties can be understood as formal differences (P−Q) of convex polyhedra in the character lattice. We show how it is possible to use this language for understanding the cohomology of L by studying the set-theoretic difference (Q\P). Moreover, when interpreting these cohomology groups as certain Ext-groups, we demonstrate how the approach via (Q\P) leads to a direct description of the associated extensions. The first part is joint work with Jarek Buczinski, Lars Kastner, David Ploog, and Anna-Lena Winz; the second is joint work with Amelie Flatt and Lutz Hille.
|27.01.2021||Sönke Rollenske (Marburg): Stratifications in the moduli space of stable surfaces |
The Gieseker moduli space of surfaces of general type admits a modular compactification, the moduli space of stable surfaces. Our knowledge about the "new" surfaces in the boundary is still limited and I will discuss different possibilities to organise them, in particular a Hodge-theoretic approach proposed by Green, Griffiths, Laza, and Robles. Everything will be illustrated with examples and many pictures. This is based on joint work with B. Anthes, M. Franciosi, R. Pardini.
|03.02.2021||Stefan Kebekus (Freiburg): Brauer-Manin obstruction on a simply connected fourfold and a Mordell theorem in the orbifold setting |
Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's étale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups. In joint work with Jorge Pereira (IMPA) and Arne Smeets (Nijmegen), we construct simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new diophantine result of independent interest: a Mordell-type theorem for Campana's "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.
|17.02.2021||Ann Sophie Cenkel (UdS): Borsuk-Ulam Theorem and its applications, II|
|29.04.2020||Frank-Olaf Schreyer (UdS): Hyperelliptic curves, complete intersection of two quadrics and their Ulrich complexity|
|06.05.2020||Cédric Bonnafé (Montpellier): Singular K3 surfaces and complex reflection groups |
Joint work with A. Sarti. Singular K3 surfaces are the K3 surfaces with maximal Picard number, namely 20. I will explain how to construct families of K3 surfaces with big Picard number using invariants of finite complex reflection groups of rank 4, each family containing some singular ones. This extends earlier work of Barth-Sarti for two reasons: firstly, we obtain many more examples by considering all reflection groups of rank 4 and secondly, our proofs involve more theory of complex reflection groups and avoid as much as possible (but not completely) a case-by-case analysis.
|13.05.2020||Calum Spicer (King's College London): Mori theory and foliations |
We will explain some of the ideas behind the study of the birational geometry of foliations, as well as indicating some recent progress in the case of codimension one foliations on threefolds. We will suggest some relations between the study of the birational geometry of foliations and the algebraicity of leaves of foliations. Features joint work with P. Cascini and R. Svaldi.
|27.05.2020||Niklas Müller (UdS): Vanishing theorems in algebraic geometry |
Sascha Blug (UdS): Gonality and relative canonical syzygies of curves on toric surfaces
|10.06.2020||Fanjun Meng (Northwestern): Pushforwards of klt pairs under morphisms to abelian varieties |
In this talk, I will discuss some positivity properties of pushforwards of klt pairs under morphisms to abelian varieties, which include global generation (after pullback by an isogeny), Chen-Jiang decomposition and some other related ones. These extend previous results about pushforwards of pluricanonical bundles of smooth projective varieties to the singular setting. I will also apply them to some effective results.
|01.07.2020||Andreas Höring (Nice): A nonvanishing conjecture for cotangent bundle |
Let X be a smooth complex projective manifold, and let KX be its canonical bundle. If X is not covered by rational curves (or equivalently KX is pseudoeffective), the nonvanishing conjecture claims that some positive multiple of the canonical bundle has a non-zero global section. In this talk I will discuss an analogous conjecture for the cotangent bundle ΩX: assume that ΩX pseudoeffective (I will give the definition in the talk). Then some symmetric power SmΩX has a non-zero global section. We will see that this conjecture is quite non-trivial, even for surfaces. This is joint work with Thomas Peternell.
|26.08.2020||Niklas Müller (UdS): Vanishing theorems in algebraic geometry, II |
Sascha Blug (UdS): Gonality and relative canonical syzygies of curves on toric surfaces, II
|23.10.2019||Vladimir Lazić (UdS): Abundance for uniruled pairs which are not rationally connected|
|06.11.2019||Patrick Graf (Bayreuth): Reflexive differential forms in positive characteristic |
Given a differential form on the smooth locus of a normal variety defined over a field of positive characteristic, we discuss under what conditions it extends to a resolution of singularities (possibly with logarithmic poles). Our main result works for log canonical surface pairs over a perfect field of characteristic at least seven. We also give a number of examples showing that our results are sharp in the surface case, and that they fail in higher dimensions.
Giovanni Staglianò (Catania): Kuznetsov's Conjecture and rationality of cubic fourfolds
We recall the main conjectures about the important classical problem (still unsolved) on the rationality of smooth cubic hypersurfaces in P5, cubic fourfolds for short, and we present recent contributions in favour of these conjectures. We will also briefly illustrate similar conjectures and results for Gushel-Mukai fourfolds, that is for smooth quadric hypersurfaces in del Pezzo fivefolds. The talk is based on some joint works with Francesco Russo and the recent collaboration with Michael Hoff.
|13.11.2019||Eileen Oberringer (UdS): Riemann-Roch theorem and applications |
Lisa Karst (UdS): Bézout's theorem and applications
Igor Schlegel (UdS): Dimension theory and applications
|08.01.2020||Andreas Knutsen (Bergen): Moduli of polarized Enriques surfaces |
Moduli spaces of polarized Enriques surfaces have several components, even if one fixes the degree of the polarization. I will present some results concerning how to determine the various irreducible components and in some cases their unirationality and uniruledness.
|15.01.2020||Frank Gounelas (TU München): Curves on K3 surfaces |
Bogomolov and Mumford proved that every projective K3 surface contains a rational curve. Since then, a lot of progress has been made by Bogomolov, Chen, Hassett, Li, Liedtke, Tschinkel and others, towards the stronger statement that any such surface in fact contains infinitely many rational curves. In this talk I will present joint work with Xi Chen and Christian Liedtke completing the remaining cases of this conjecture in characteristic zero, reproving some of the main previously known cases more conceptually and extending the result to arbitrary genus.
|29.01.2020||Enrica Floris (Poitiers): On Mori fibre spaces of dimension 4 and their automorphism group |
In this talk we will explain how the study of Mori fibre spaces with an action of an algebraic connected group is related to the study of maximal connected subgroups of the Cremona group. We will present some examples of Mori fibre spaces of dimension 4 fibred onto the projective line. This is a work in progress joint with Jérémy Blanc.
|05.02.2020||Yongqiang Zhao (Westlake): Scrollar syzygies and Galois representations |
The theory of scrollar syzygy resolutions was introduced by Schreyer in his work on Green's conjecture for canonical curves. Apart from its applications to syzygy theory, it is also closely related to the parametrization theory of curves having small gonality, which has important applications to the theory of Hurwitz spaces. A priori, it has nothing to do with representation theory. In this talk, however, we will discuss some recent observations on its connections with Galois representation theory. We will give a (conjectural) complete description of all syzygy bundles through representation theory.
Wouter Castryck (Leuven): Recovering a secret ideal class modulo squares
The "discrete logarithm problem for class group actions" is about finding a secret ideal class [J] in the class group of an imaginary quadratic order O, upon input of two elliptic curves E, E' (over a finite field), both having endomorphism ring O, that are obtained from one another through the isogeny-wise action of [J]. This problem was proposed by Couveignes as a substitute for the classical discrete logarithm problem in elliptic curve cryptography, and has recently attracted a lot of attention, thanks to the quantum computing threat. In this talk we will introduce this problem in more detail, and we will explain how to efficiently recover [J] modulo the subgroup of squares inside the class group of O. In many cases, this breaks the decisional version of the discrete logarithm problem for class group actions. This is joint work with Jana Sotáková and Frederik Vercauteren.
|19.02.2020||Eileen Oberringer (UdS): Riemann-Roch theorem and applications, II |
Lisa Karst (UdS): Bézout's theorem and applications, II
Igor Schlegel (UdS): Dimension theory and applications, II
|10.04.2019||Alberto Cattaneo (MPI Bonn): Positive divisors and automorphisms of hyperkähler manifolds |
The movable cone of a projective hyperkähler manifold admits a locally polyhedral wall-and-chamber decomposition which encodes information on the birational models of the manifold. In the case of moduli spaces of Bridgeland-stable objects on K3 surfaces, Bayer and Macrì provided a lattice-theoretical description of the walls in this decomposition, which allows for explicit computations. We will show how to apply these results to obtain a purely arithmetic classification of the automorphism group of Hilbert schemes of points on a generic projective K3 surface.
|17.04.2019||Davide Veniani (Mainz): The number of Enriques surfaces covered by a K3 surface |
Given the transcendental lattice of a complex K3 surface X, I will give a formula for the number of non-isomorphic Enriques surfaces covered by X. In particular, I will describe the two 'most algebraic' Enriques surfaces covered by the singular K3 surface of discriminant 7 (j.w. Ichiro Shimada). Finally, I will explain how the search for K3 surfaces with no Enriques quotients turns into a question about integral quadratic forms representing 1, and how to answer it (j.w. Simon Brandhorst).
|08.05.2019||Sławomir Rams (Cracow): On lines on surfaces of general type |
I will present bounds on the number of lines on a smooth degree d surface in three-dimensional projective space for d=5 (joint work with M. Schuett) and d>5 (joint with T. Bauer).
|15.05.2019||Ichiro Shimada (Hiroshima): The automorphism groups of Enriques surfaces covered by a Jacobian Kummer surface|
|22.05.2019||Nikolaos Tsakanikas (UdS): On minimal models|
|28.05.2019||Francesco Galuppi (MPI Leipzig): The rough Veronese variety |
We study signature tensors of paths from an algebraic geometric viewpoint. The signatures of a given class of paths parametrize a variety inside the space of tensors. These "signature varieties" provide both new tools to investigate paths and new challenging questions about their behavior. In this talk I will focus on the class of rough paths. They play a central role in stochastic analysis, and their signature variety has peculiar geometric properties, showing surprising analogies with the classical Veronese variety. We show that this so-called Rough Veronese is toric. This makes it much easier to study it and to perform explicit computations.
|05.06.2019||Yeongrak Kim (UdS): On the Borisov-Nuer conjecture|
|12.06.2019||Rosemary Taylor (Warwick): On explicit constructions of codimension 4 Fano 3-folds |
Fano 3-folds with terminal singularities are a key family produced by the minimal model program, but their classification remains an open problem. Often research into these varieties takes place on their Gorenstein anticanonical rings. A famous classification of Gorenstein rings in codimensions 1, 2 and 3 allows us to classify Fano 3-folds in these codimensions; however, codimension 4 Fano 3-folds remain mysterious. In this talk I will introduce "unprojections". Unprojections are a graded ring method which construct "large" Gorenstein rings from "smaller" and in practice act as a good substitute for classification and structure theories. In particular, I will show how to apply this method to the case of Fano 3-folds and prove the existence of 16 new codimension 4 Fano 3-folds.
|26.06.2019||Isabel Stenger (UdS): Constructing numerical Godeaux surfaces|
|17.07.2019||Sijong Kwak (Daejeon): The Betti tables of higher secant varieties |
In this talk, I'd like to consider the defining equations of higher secant varieties and the structure of the Betti tables. There are very similar structures as in the quadratic world. I'd like to call this phenomenon "Matryoshka structure" which I discussed with Fyodor Zak long time ago.
|17.10.2018||Vladimir Lazić (UdS): On Generalised Abundance|
|24.10.2018||Vladimir Lazić (UdS): On Generalised Abundance, II|
|07.11.2018||Sascha Blug (UdS): Divisors and line bundles on hyperelliptic curves|
|14.10.2018||Vladimir Lazić (UdS): On the B-Semiampleness Conjecture|
|28.11.2018||Roberto Svaldi (Cambridge): On the boundedness of Calabi-Yau varieties in low dimension |
I will discuss new results towards the birational boundedness of low-dimensional elliptic Calabi-Yau varieties, joint work with Gabriele Di Cerbo and work in progress with Caucher Birkar and Di Cerbo. Recent work in the minimal model program suggests that pairs with trivial log canonical class should satisfy some boundedness properties. I will show that 4-dimensional Calabi-Yau pairs which are not birational to a product are indeed log birationally bounded. This implies birational boundedness of elliptically fibered Calabi-Yau manifolds with a section, in dimension up to 5. If time allows, I will also try to discuss a first approach towards boundedness of rationally connected CY varieties in low dimension (joint with G. Di Cerbo, W. Chen, J. Han and C. Jiang).
|19.12.2018||Simon Brandhorst (UdS): On the dynamical spectrum of complex surfaces|
|09.01.2019||Chiara Camere (Milano/MPI Bonn): Twisted sheaves on K3 surfaces, Verra fourfolds and non-symplectic involutions |
The object of this talk is the construction of a family of hyperkähler fourfolds of K3-type producing an example of the last missing case in the classification of non symplectic involutions. We will see that these varieties can be described geometrically in two different ways, as moduli spaces of twisted sheaves on K3 surfaces and as double covers of EPW quartics associated to Verra fourfolds. This is joint work with G. Kapustka, M. Kapustka and G. Mongardi.
|16.01.2019||Fabio Bernasconi (Imperial College London): On log del Pezzo surfaces over imperfect fields and Mori fibre spaces |
Fibrations play a key role in the classification problems of algebraic varieties. While in characteristic zero, the general fibre of a morphism between smooth varieties is still smooth, this is no longer true in general over fields of positive characteristic (where the classical examples are quasi-elliptic fibrations). However, one can hope to bound such a bad behaviour to small primes if the generic fibre has special global properties. In this talk, I will discuss the case log del Pezzo surfaces over imperfect fields, with a particular emphasis on their cohomologies and Picard groups. This has interesting consequences to the study of threefold Mori fibre spaces onto a curve. This is a joint work with H. Tanaka.
|23.01.2019||Susanna Zimmermann (Angers): Relations between Sarkisov links |
A result by Hacon and McKernan states that the groupoid of birational maps between Mori fibre spaces is generated by Sarkisov links. We revisit and refine a result of Kaloghiros, which presents a set of generating relations between Sarkisov links, and give a closer look at those which include Sarkisov links between conic bundles. This is joint work with Jérémy Blanc and Stéphane Lamy.
|26.04.2018||Zsolt Patakfalvi (EPF Lausanne): Positivity of the Chow-Mumford line bundle for families of K-stable Q-Fano varieties |
The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of polarized varieties, in particular on the base of families of Q-Fano varieties (that is, Fano varieties with klt singularities). It is conjectured that it yields a polarization on the conjectured moduli space of K-semi-stable Q-Fano varieties. This boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable Q-Fano fibers. I present a joint work with Giulio Codogni where we prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming K-stability only for very general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. I also present a birational geometry application to the classification of Fano varieties.
|03.05.2018||Christian Lehn (TU Chemnitz): Coisotropic subvarieties in holomorphic symplectic manifolds |
The existence of coisotropic subvarieties in general projective irreducible symplectic varieties was conjectured by C. Voisin and is related to the structure of the Chow ring of such a variety and in particular to zero cycles. We present some methods to prove general existence results and apply them on special irreducible symplectic varieties. This is joint work with Gianluca Pacienza.
|17.05.2018||Stefan Schreieder (LMU München): Stably irrational hypersurfaces of small slopes |
We show that a very general complex projective hypersurface of dimension N and degree at least log2N+2 is not stably rational. The same statement holds over any uncountable field of characteristic p>>N.
|24.05.2018||Christian Liedtke (TU München): A Néron-Ogg-Shafarevich criterion for K3 Surfaces |
Given a family of smooth and projective manifolds over a pointed disk, one may ask whether this family has good reduction, that is, whether this family can be extended to a smooth family over the whole disk. A necessary condition for this is that the monodromy actions on cohomology, that is, the actions of the fundamental group of the pointed disk on the singular cohomology groups of a general fiber, is trivial. But does the converse hold? First, we generalise the setup as follows: let R be a complete local ring with field of fractions K and residue field k (Spec K generalises the pointed disk and Spec R generalises the whole disk), and let X be a smooth and projective variety over K. In this setup, good reduction translates into finding a smooth and proper scheme over Spec R with generic fiber X. Moreover, trivial monodromy translates into the action of the absolute Galois group of K on all l-adic cohomology groups H*(X,Ql) being unramified (or crystalline if l=char(k)).
For Abelian varieties, it is classical known by a theorem of Serre and Tate (and already established by Néron, Ogg, and Shafaravich for elliptic curves) that good reduction is equivalent to having unramified/crystalline Galois-actions on H1.
In my talk, I will first introduce the above notions and then, I will talk on joint work with Matsumoto, Chiarellotto, and Lazda, where we study K3 surfaces over K and ask whether good reduction is equivalent to having unramfied/crystalline Galois-actions on H2. It turns out that this is almost true, but that the right analog of a Néron-Ogg-Shafarevich criterion for K3 surfaces is rather subtle.
|21.06.2018||Hanieh Keneshlou (UdS): The unirational component of moduli stacks of 6 gonal genus 11 curves|
|28.06.2018||Cinzia Casagrande (Torino): Fano 4-folds with rational fibrations |
Smooth, complex Fano 4-folds are not classified, and we still lack a good understanding of their general properties. In the talk we will focus on Fano 4-folds with large second Betti number b2, studied via birational geometry and the detailed study of their contractions and rational contractions. We recall that a contraction is a morphism with connected fibers onto a normal projective variety, and a rational contraction is given by a sequence of flips followed by a (regular) contraction.
The main result that we want to present is the following: let X be a Fano 4-fold having a rational contraction X→Y of fiber type (with dim Y > 0). Then either X is a product of surfaces, or b2(X) is at most 17, or Y is P1 or P2.
|05.07.2018||Alessandra Sarti (Poitiers): Involutions of the Hilbert scheme of two points on a K3 surface |
I will show how to use K3 surfaces with special geometry to construct involutions on their Hilbert scheme of two points. Indeed by using Torelli theorem one can show the existence of automorphisms of the Hilbert scheme, but it is a difficult problem to give an explicit realization. This is a joint work with S. Boissière, A. Cattaneo and D. Markushevich.
|12.07.2018||Michael Hoff (UdS): K3 surfaces of small genus with many elliptic pencils|
|19.10.2017||Pagona Koulakidou (UdS): Divisors as fibres of morphisms |
Nikolaos Tsakanikas (UdS): On varieties birational to abelian varieties
|07.12.2017||George H. Hitching (OsloMet): Tangent cones to generalised theta divisors and generic injectivity of the theta map |
Let C be a Petri general curve of genus g. The tangent cones of the Riemann theta divisor on Picg-1(C) have been used in various ways by Kempf and Schreyer and by Ciliberto and Sernesi to give new proofs of Torelli's theorem. We use a related approach to study the generalised theta divisor D(V) of a semistable bundle V over C of rank r and integral slope. For large enough g, we show how a sufficiently general such V can be reconstructed from the tangent cone of D(V) at a suitable singular point. We use this to give a constructive proof and a sharpening of Brivio and Verra's theorem that the theta map from the moduli space of semistable bundles of rank r and trivial determinant to the projective space |rΘ| is generically injective for large values of g. (Joint work with Michael Hoff)
|18.01.2018||Gianluca Pacienza (Nancy): Density of Noether-Lefschetz loci of irreducible holomorphic symplectic varieties and applications |
We will try to illustrate how useful such density results can be by presenting several (old and new) applications to: the existence of rational curves on projective IHS varieties, the study of relevant cones of divisors, the study of lagrangian fibrations and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is isomorphic to to an Hilbert scheme of 2 points on a K3 surface. We also discuss Voisin's conjecture on the existence of coisotropic subvarieties on IHS varieties and relate it to a stronger statement on Noether-Lefschetz loci in their moduli spaces.
|25.01.2018||Andrea Fanelli (Düsseldorf): Del Pezzo fibrations in positive characteristic |
In this talk, I will discuss some pathologies for the generic fibre of del Pezzo fibrations in characteristic p>0, motivated by the recent developments of the MMP in positive characteristic. The main application of the joint work with Stefan Schröer concerns 3-dimensional Mori fibre spaces.
|01.02.2018||Luca Tasin (Bonn): A non-vanishing result for weighted complete intersections |
Let X be a smooth (or mildly singular) projective variety and let H be an ample line bundle on X. Kawamata conjectured that if H-KX is ample, then the linear system |H| is not empty. I will explain that the conjecture holds true for weighted complete intersections which are Fano or Calabi-Yau, relating it with the Frobenius coin problem. This is based on a joint work with M. Pizzato and T. Sano.
|18.05.2017||Christian Bopp (UdS): Moduli of lattice polarized K3 surfaces via relative canonical resolutions|
|01.06.2017||Sascha Blug (UdS): Die Picard Gruppe von hyperelliptischen Kurven |
Jonas Baltes (UdS): Der Riemannsche Abbildungssatz und Metriken konstanter Krümmung auf Riemannschen Flächen
|08.06.2017||Janik Schug (UdS): Puiseux-Reihen |
Christian Ikenmeyer (UdS): Formula size, iterated matrix multiplication, and algebraic geometry
|29.06.2017||Andreas L. Knutsen (Bergen): Brill-Noether theory of curves on abelian surfaces |
The Brill-Noether theory of curves on K3 surfaces is well understood. Until recently, quite little has been known for curves on abelian surfaces. In the talk I will present some recent results obtained with M. Lelli-Chiesa and G. Mongardi.
In particular, we show that the general curve in the linear system |L| on a general primitively polarized abelian surface (S,L) is Brill-Noether general, as in the K3 case. However, contrary to the K3 case, there are smooth curves in |L|possessing "unexpected" linear series, that is, with negative Brill-Noether number. As an application, we obtain the existence of components of special Brill-Noether loci of the expected dimension in the moduli space of curves.
Thomas Peternell (Bayreuth): Descent of numerically flat vector bundles and singular ball quotients
In my talk I will explain recent results with Greb, Kebekus and Taji concerning the uniformization of klt spaces whose (orbifold) Chern classes are extremal in the sense that they satisfy the Miyaoka-Yau equality.
|06.07.2017||Hanieh Keneshlou (UdS): On Accola's genus bound for algebraic curves |
Frank-Olaf Schreyer (UdS): Horrocks splitting on products of projective spaces
|03.08.2017||Michael Kemeny (Stanford): The Prym-Green conjecture for curves of odd genus |
We will present a proof of the Prym-Green conjecture on the resolution of a paracanonical curve of odd genus and arbitrary torsion level. The proof proceeds by using curves on ruled surfaces over an elliptic curve. These surfaces naturally arise as desingularizations of limiting K3 surfaces with elliptic singularities, and come up in Arbarello-Bruno-Sernesi's study of the Wahl map and deformations of the cone. They have the downside of being irregular, which makes the study of syzygies more complicated than for K3s, but on the upside they allow for inductive arguments on the genus of the curve, which is not possible for a K3. Joint with Gabi Farkas.
|09.08.2017||Yeongrak Kim (UdS): Ulrich bundles on the intersection of two 4-dimensional quadrics |
A coherent sheaf F on a projective variety X is Ulrich if its pushforward by a finite degree map is trivial. Since they naturally appear in several different theories, the study of Ulrich bundles becomes important. In this talk, I will discuss two different approaches to construct Ulrich bundles on the intersection of two 4-dimensional quadrics: via Serre correspondence and via derived categories. I will also briefly explain a connection between generalized theta series. This is a joint work with Y. Cho and K.-S.Lee.
|24.08.2017||Jonas Baltes (UdS): Der Riemannsche Abbildungssatz und Metriken konstanter Krümmung auf Riemannschen Flächen (Teil II)|
|05.10.2017||Janik Schug (UdS): Puiseuxreihen und ihre Anwendungen|