Oberseminar Algebraische Geometrie

We meet regularly on Wednesdays in the building E2 4, room SR 10, starting at 2pm.

NEWS (Sommersemester 2022)
We meet regularly on Wednesdays in SR10 or online in Zoom, starting at 2pm.

Sommersemester 2022

Vladimir Lazić (UdS): The Nonvanishing problem for varieties with nef first Chern class
Let X be a threefold with mild singularities such that c1(X) is nef. In this talk I will present a very recent proof that then the numerical class of c1(X) is effective. This result (joint work with Thomas Peternell, Nikolaos Tsakanikas and Zhixin Xie) is the positive curvature counterpart of the famous result of Miyaoka from the 1980s, which showed that –c1(X) is effective as soon as it is nef.


Sokratis Zikas (Basel): Rigid birational involution of projective space and cubic 3-folds
The Cremona group Crn(k) is the group of birational transformations of the projective n-space over a field k. The study of these groups dates back to the 19th century with some of the central questions still being open. In this talk, using modern techniques based on the Minimal Model Program, I will explain how to construct families of birational involutions on the projective 3-space which do not fit in an elementary relation of Sarkisov links. Using these involutions, we can endow Cr3(C) with a free product structure. As corollaries, we effectively reprove its non-simplicity as well as answer some question regarding its group of automorphisms. Counterpart results hold for the group of birational transformations of a smooth cubic 3-fold. 

04.05.2022Quentin 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.
11.05.2022Frank-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 YPn+1 of dimension dim X+1, which is not a cone, is an extension of a variety XPn if there exists an hyperplane HPn+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 CP4.
at 4pm!

Jingjun Han (Fudan): Birational boundedness of rationally connected log Calabi-Yau pairs with fixed index
In this talk, I will introduce McKernan-Prokhorov's conjecture on the boundedness of rationally connected Calabi-Yau varieties. This conjecture is a natural generalization of the Borisov– Alexeev–Borisov (BAB) Conjecture which was solved by Birkar. I will show that the set of rationally connected projective varieties X of a fixed dimension such that (X,B) is klt, and -m(KX+B) is Cartier and nef for some fixed positive integer m, is bounded modulo flops. This is a joint work with Chen Jiang.

at 2:30pm!

Fulvio Gesmundo (UdS): Lower bounds for algebraic branching programs via intersection theory
Algebraic branching programs (ABP) define one of the algebraic versions of the class of problems which can be solved in polynomial time. In this seminar, I will explain how to characterize ABPs via restrictions of a special class of polynomials and how geometric methods allow us to determine lower bounds in this class. In particular, I will focus on recent work with Ghosal, Ikenmeyer and Lysikov, where we use intersection theoretic properties of an associated hypersurface to give lower bounds on the ABP complexity.

29.06.2022Robin Lahni (UdS, Bachelorseminar): Numerical criteria for ampleness and nefness
Lukas Braun (Freiburg): Reductive quotients of klt varieties

In this talk, I will explain the proof of the recent result, obtained together with Daniel Greb, Kevin Langlois, and Joaquin Moraga, that reductive quotients of klt type varieties are of klt type. If time permits, I will also discuss several applications of our result, e.g. on quotients of Fano type varieties, good moduli spaces, and collapsing of homogeneous bundles.
at 11am!
Shin-ichi Matsumura (Tohoku/Bayreuth): On projective manifolds with pseudo-effective tangent bundle
In this talk, I will introduce a structure theorem of a projective manifold with pseudo-effective tangent bundle, comparing to the structure theorem for nef cotangent bundle by Demailly-Peternell-Schneider. Specifically, I show that such a manifold admits a smooth fibration onto a flat projective manifold whose general fiber is rationally connected. The proof is based on the theory of foliations, but an analog of the structure theorem is proved from the viewpoint of MMP. I would like to explain a comparison of the viewpoints of foliations and MMP. This talk is partially based on joint work with Genki Hosono and Masataka Iwai.
at 10am!
Jie Liu (Beijing): Bounding algebraic ranks of Fano foliations
One of the central problems in the theory of holomorphic foliations on projective varieties is to find conditions that guarantee the existence of algebraic leaves and the notion of algebraic rank was introduced to measure the algebraicity of leaves. In this talk, I will talk about how we can get a lower bound for algebraic ranks of Fano foliations using global and local invariants related to the positivity of anti-canonical divisors.
13.07.2022Joachim Jelisiejew (Warsaw): Geometry of minimal border rank tensors
Concice tensors in CmCmCm which have border rank m are classically important in complexity theory. In the talk I will present a new approach to these tensors, centered around the notion of 111-algebra. It yields both new results, like set-theoretic equations for m=5, as well as a new perspective and abundance of open questions, ranging from computational, to birational, to deformation theoretic. This is a joint work with J.M. Landsberg and Arpan Pal, building on the work of Buczyńska-Buczyński.
at 2pm!
Robin Lahni (UdS, Bachelorseminar): Numerical criteria for ampleness and nefness, II
Wintersemester 2021/22

Nikolaos Tsakanikas (UdS): Termination of flips for pseudo-effective 4-folds
The Minimal Model Program (MMP) plays a fundamental role in the classification theory of projective varieties. One of the main open problems of the MMP in higher dimensions is the termination of flips conjecture, which predicts that any sequence of flips terminates. In this talk I will sketch the proof of the termination of flips conjecture for pseudo-effective log canonical generalized pairs of dimension 4. This is joint work with Guodu Chen.


Zhixin Xie (UdS): Rationally connected threefolds with nef anticanonical divisor
Complex projective varieties with nef anticanonical divisor appear as natural generalisations of Fano varieties. Fano manifolds are classified up to dimension three and many of their properties are well studied. However, the classification of manifolds with nef anticanonical divisor is more complicated as new phenomena arise and many results for Fano manifolds no longer hold for this class of varieties. In this talk, we will first look at some examples in dimension two. Then we will discuss some properties of rationally connected threefolds with nef anticanonical divisor. We will consider the case where the anticanonical divisor is not semi-ample, and will give a classification result in this case.

10.11.2021Giacomo 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.
17.11.2021Stefano 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.
24.11.2021Giovanni 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
We will present recent advances in the field of singular varieties with trivial canonical class obtained in joint work with Bakker and Guenancia building on work by many others. 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.

at 9am!
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
Lesieutre showed that various definitions of the numerical dimension do not coincide for a pseudoeffective R-divisor on a specific complete intersection Calabi-Yau threefold. We show that Lesieutre's example is not a sporadic one and generalize his method of proof. This is joint work with Michael Hoff.

Sommersemester 2021

Yvonne Neuy (UdS, Bachelorseminar): Factorization of polynomials
Sven Manthe (UdS, Bachelorseminar): Generation of local integral unitary groups


Hans-Christian Graf von Bothmer (Hamburg): Rigid, not infinitesimally rigid surfaces with ample canonical bundle
It was a long standing problem of Morrow and Kodaira whether there are compact complex manifolds X with Def(X) a non reduced point. The first examples answering this question in the affirmative were given by Bauer and Pignatelli in 2018. These are certain surfaces of general type that have nodal canonical models. These canonical models are rigid and infinitesimally rigid, while their desingularizations are still rigid, but not infinitesimally rigid anymore. One can therefore ask, if this situation is typical for rigid, not infinitesimally rigid surfaces of general type, or if it is possible to have examples with smooth canonical models. We answer this question also in the affirmative by constructing such a surface via line arrangements and abelian covers.

14.07.2021Firoozeh 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.

at 10am!

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.
28.07.2021Yvonne Neuy (UdS, Bachelorseminar): Factorization of polynomials, II
Wintersemester 2020/21
04.11.2020Yeongrak Kim (UdS): Ulrich bundles on cubic fourfolds
11.11.2020Andreas 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.
02.12.2020Ann Sophie Cenkel (UdS, Bachelorseminar): Borsuk-Ulam Theorem and its applications
09.12.2020Edgar 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.
at 4pm!
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 (PQ) 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. 
27.01.2021Sönke 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.
03.02.2021Stefan 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.
17.02.2021Ann Sophie Cenkel (UdS, Bachelorseminar): Borsuk-Ulam Theorem and its applications, II
Sommersemester 2020
29.04.2020Frank-Olaf Schreyer (UdS): Hyperelliptic curves, complete intersection of two quadrics and their Ulrich complexity
06.05.2020Cédric 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.
13.05.2020Calum 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.
27.05.2020Niklas Müller (UdS, Bachelorseminar): Vanishing theorems in algebraic geometry 
Sascha Blug (UdS, Masterseminar): Gonality and relative canonical syzygies of curves on toric surfaces
10.06.2020Fanjun 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. 
01.07.2020Andreas 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.
26.08.2020Niklas Müller (UdS, Bachelorseminar): Vanishing theorems in algebraic geometry, II 
Sascha Blug (UdS, Masterseminar): Gonality and relative canonical syzygies of curves on toric surfaces, II
Wintersemester 2019/20
23.10.2019Vladimir Lazić (UdS): Abundance for uniruled pairs which are not rationally connected
06.11.2019Patrick 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.
13.11.2019Eileen Oberringer (UdS, Bachelorseminar): Riemann-Roch theorem and applications 
Lisa Karst (UdS, Bachelorseminar): Bézout's theorem and applications 
Igor Schlegel (UdS, Bachelorseminar): Dimension theory and applications
08.01.2020Andreas 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.
15.01.2020Frank 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.
29.01.2020Enrica 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.
05.02.2020Yongqiang 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 EE' (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.
19.02.2020Eileen Oberringer (UdS, Bachelorseminar): Riemann-Roch theorem and applications, II 
Lisa Karst (UdS, Bachelorseminar): Bézout's theorem and applications, II 
Igor Schlegel (UdS, Bachelorseminar): Dimension theory and applications, II
Sommersemester 2019
10.04.2019Alberto 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.
17.04.2019Davide 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). 
08.05.2019Sławomir 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).
15.05.2019Ichiro Shimada (Hiroshima): The automorphism groups of Enriques surfaces covered by a Jacobian Kummer surface
22.05.2019Nikolaos Tsakanikas (UdS): On minimal models
28.05.2019Francesco 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.
05.06.2019Yeongrak Kim (UdS): On the Borisov-Nuer conjecture
12.06.2019Rosemary 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. 
26.06.2019Isabel Stenger (UdS): Constructing numerical Godeaux surfaces
17.07.2019Sijong 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.
Wintersemester 2018/19
17.10.2018Vladimir Lazić (UdS): On Generalised Abundance 
24.10.2018Vladimir Lazić (UdS): On Generalised Abundance, II 
07.11.2018Sascha Blug (UdS, Bachelorseminar): Divisors and line bundles on hyperelliptic curves 
14.10.2018Vladimir Lazić (UdS): On the B-Semiampleness Conjecture 
28.11.2018Roberto 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).
19.12.2018Simon Brandhorst (UdS): On the dynamical spectrum of complex surfaces 
09.01.2019Chiara 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[2]-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.
16.01.2019Fabio 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.
23.01.2019Susanna 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.
Sommersemester 2018
26.04.2018Zsolt 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.
03.05.2018Christian 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.
17.05.2018Stefan 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.
24.05.2018Christian 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.
21.06.2018Hanieh Keneshlou (UdS): The unirational component of moduli stacks of 6 gonal genus 11 curves
28.06.2018Cinzia 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.
05.07.2018Alessandra 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.
12.07.2018Michael Hoff (UdS): K3 surfaces of small genus with many elliptic pencils
Wintersemester 2017/18
19.10.2017Pagona Koulakidou (UdS): Divisors as fibres of morphisms 
Nikolaos Tsakanikas (UdS): On varieties birational to abelian varieties
07.12.2017George 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)
18.01.2018Gianluca 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.
25.01.2018Andrea 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.
01.02.2018Luca 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.
Sommersemester 2017
18.05.2017Christian Bopp (UdS): Moduli of lattice polarized K3 surfaces via relative canonical resolutions
01.06.2017Sascha Blug (UdS, Bachelorseminar): Die Picard Gruppe von hyperelliptischen Kurven 
Jonas Baltes (UdS, Bachelorseminar): Der Riemannsche Abbildungssatz und Metriken konstanter Krümmung auf Riemannschen Flächen
08.06.2017Janik Schug (UdS, Bachelorseminar): Puiseux-Reihen 
Christian Ikenmeyer (UdS): Formula size, iterated matrix multiplication, and algebraic geometry
29.06.2017Andreas 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.
06.07.2017Hanieh Keneshlou (UdS): On Accola's genus bound for algebraic curves 
Frank-Olaf Schreyer (UdS): Horrocks splitting on products of projective spaces
03.08.2017Michael 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.
09.08.2017Yeongrak 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.
24.08.2017Jonas Baltes (UdS, Bachelorseminar): Der Riemannsche Abbildungssatz und Metriken konstanter Krümmung auf Riemannschen Flächen (Teil II)
05.10.2017Janik Schug (UdS, Bachelorseminar): Puiseuxreihen und ihre Anwendungen


Fachrichtung Mathematik
Campus, Gebäude E2 4
Universität des Saarlandes
66123 Saarbrücken