Oberseminar Algebraische Geometrie

Zhan Li (SUSTech): On the relative Morrison-Kawamata cone conjecture for Calabi-Yau fiber spaces
A fibration with a relatively trivial canonical divisor is called a Calabi-Yau fiber space. The Morrison-Kawamata cone conjecture relates the birational geometry of a Calabi-Yau fiber space to the convex geometry of a movable cone. Assuming the minimal model program conjectures, we show that the cone conjecture of the generic fiber implies the cone conjecture of the Calabi-Yau fiber space. As an application, the finiteness of minimal models of a Calabi-Yau fiber space in relative dimensions less or equal to 2 is established. This work is partially joint with Hang Zhao.
Omprokash Das (Tata Institute): Recent developments in the minimal model program for Kähler varieties
Some of the fundamental theorems of MMP, such as Mori’s Bend-Break, Base-point free theorem, etc. fail on compact Kahler manifolds. Despite these failures, in a series of three pairs in 2015-16, Campana, Peternell and Höring established that the MMP holds for compact Kahler threefolds with terminal singularities. In this talk I will explains some of the main tools and techniques developed by these authors. Recently we have extended these results to DLT pairs in dim 3 and some partial cases in dim 4, and also introduced an inductive approach to the contraction problem. In this talk I will survey these results and explain some of our techniques. These are part of joint work with Christopher Hacon.

Annalisa Grossi (Paris-Saclay): Birational equivalence and deformation equivalence for Hyperkähler manifolds
Huybrechts proved that if X and Y are birational HK manifolds they have to be deformation equivalent, but this implication is far from being an equivalence. It is then natural to ask which additional assumptions are needed to get that two HK manifolds of the same deformation type are birational. In this talk I will give a gentle introduction on lattice theory for HK manifolds, then I will provide a lattice-theoretic criterion to determine when two HK manifolds of OG10 type are birational (joint work with C. Felisetti and F. Giovenzana), and I will show an application to the Li-Pertusi-Zhao variety.

Zheng Xu (AMSS Beijing): On the abundance conjecture for threefolds in positive characteristic
Over the past decade, the Minimal Model Program (MMP) has been largely established for threefolds over perfect fields of characteristic >3. A central conjecture remaining is the abundance conjecture, which predicts that if (X,B) is a projective log canonical threefold pair over an algebraically closed field of characteristic >3 and K_X+B is nef, then K_X+B is semiample. In this talk, I will discuss some recent results towards this conjecture. For example, I will explain my proof of this conjecture in the case when the numerical dimension of K_X+B equals to 2. 

Consider the following two popular problems from combinatorial topology:
1. Given a piecewise linear manifold M, can we construct a triangulation of M?
2. Given two triangulations, do they represent the same piecewise linear manifold?
In dimension two, both of these problems are trivial. In dimension three, both have practical solutions (in many cases). In dimension four, however, they can become interesting and difficult.
I will discuss both problems by means of the K3 surface, a simply connected piecewise linear four-manifold: Two triangulations exist of 4-manifolds homeomorphic to the K3 surface. One of them is piecewise linear homeomorphic (diffeomorphic) to the K3 surface. The other one is highly symmetric with the smallest number of simplices – but its piecewise linear type is still unknown. 
I will present the methods used to construct both triangulations, and a list of attempts to prove their piecewise linear equivalence via local modifications, such as bi-stellar moves.

Stevell Muller (UdS): On symplectic birational transformations of OG10-type hyperkähler manifolds via cubic fourfolds
We know thanks to the work of L. Giovenzana, A. Grossi, C. Onorati and D. Veniani that OG10-type hyperkähler manifolds do not admit any non-trivial finite symplectic automorphisms. What about non-regular symplectic birational transformations? Given a cubic fourfold V, one can construct a hyperkähler manifold XV of OG10-type following a construction of R. Laza, G. Saccà, C. Voisin. Such manifolds are known as LSV manifolds. It can be shown that any symplectic automorphism on V induces a symplectic birational transformation on XV. In a couple of works with L. Marquand, we study and classify all possible cohomological actions on the OG10-lattice which can be realised as symplectic birational transformations. By investigating further the induced action on cohomology, we exhibit a criterium to decide which of these actions can be realised as induced from a cubic fourfold on an associated LSV manifold.

Víctor González Alonso (Hannover): Infinitesimal rigidity of certain modular morphisms
The Torelli morphism maps (the isomorphism class of) a smooth complex projective curve to its polarized jacobian variety. It has been recently proved by Farb that this is the only non constant holomorphic map from the moduli space of curves to that of principally polarized abelian varieties, and Serván has recently proved a similar result for the Prym morphism. One can equivalently state these results on "global rigidity" by saying that a certain moduli space of morphisms consists of just one point. But it could still happen that this point is not reduced, or in other words, the morphism admits non-trivial infinitesimal deformations. In this talk I will present a joint work with Giulio Codogni and Sara Torelli showing that this is not the case if one considers the moduli stacks. The proof uses the Fujita decomposition of the Hodge bundle of a family of curves, and can be applied to other morphisms involving moduli of smooth curves.

12.04.2023
(Wednesday!)

Eleni Agathocleous (CISPA): Elliptic Curves of j-invariant zero and the Ideal Class Group
We will present the method of descent on elliptic curves of j-invariant zero, admitting a rational 3-isogeny, λ. We will focus on a family E_D of j-invariant zero elliptic curves that we have constructed, and their λ-isogenous curves E_D', where D' = -3D are both fundamental discriminants. We will then present some of our results related to the Selmer and Tate-Shafarevich group of these elliptic curves, and show an explicit relation between these groups, and the 3-part of the ideal class group of the underlying quadratic number fields Q(√D) and Q(√D').

David Eisenbud (Berkeley): Finite and infinite free resolutions
I will survey some of the history and motivations of the study of free resolutions, from Cayley and Hilbert through Auslander, Buchsbaum and Serre on the finite case and Tate, Gulliksen, Roos, Avramov and others on the infinite case. I'll also describe some phenomena recently discovered in the infinite case by Hai Long Dao, Michael Brown, Prashanth Sridhar and myself. 

Stavros Papadakis (Ioannina): The Hibi-Ohsugi conjecture for IDP Gorenstein Lattice Polytopes
A lattice polytope P is a convex polytope whose vertices have integer coordinates. Given a field K there is an associated commutative graded K-algebra K[P]. The polytope is called Gorenstein if K[P] is Gorenstein and IDP if for all positive integers k every point with integer coordinates of the dilation polytope kP is a sum of k points of P with integer coordinates. Counting the number of points with integer coordinates of kP for all positive integers k leads to the notion of the h*-vector of P and Ehrhart theory. In 2006 Hibi and Ohsugi conjectured that the h*-vector of an IDP Gorenstein lattice polytope is unimodal. The aim of the presentation is to discuss a recent proof of the conjecture in joint work with Karim Adiprasito, Vasiliki Petrotou and Johanna Steinmeyer.

Gebhard Martin (Bonn): On the non-degeneracy of Enriques surfaces
I will report on joint work with G. Mezzedimi and D. Veniani on the classification of Enriques surfaces S with non-degeneracy nd(S)≤3, where nd(S) is the maximal length of sequences of the form (F_1,...,F_n), where |2F_i| is a genus one pencil on S and F_i∙F_j=1-δ_ij. In particular, we prove that nd(S)≥4 for all S if char(k)≠2. As a consequence, we show that, in these characteristics, all Enriques surfaces arise via Enriques' original construction as a minimal resolution of a sextic in P³ which is non-normal along the edges of a tetrahedron and all Enriques surfaces are birational to normal quintics in P³.

02.11.2022
(online)

Xavier Roulleau (Marseille): On order 3 generalized Kummer surfaces and Fourier-Mukai partners
Let A be an abelian surface which possesses a symplectic order 3 automorphism s. The minimal resolution of the quotient of A by s is a K3 surface X called a generalised Kummer surface. A natural question (analogous to the one raised by Shioda in the case of the classical Kummer surfaces) is to understand in the case when two abelian surfaces A, B with symplectic automorphisms of order 3, give the same generalised Kummer surface, is it true that A and B isomorphic? Or else, how are they related? A first step is to prove that these surfaces A, B are Fourier-Mukai partners. This will be the topic of the talk; the tool we use are of lattice theoretic nature. Joint work with A. Sarti.

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 Cr_n(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. 

Jingjun Han (Fudan): Birational boundedness of rationally connected log Calabi-Yau pairs with fixed index
In this talk, I will introduce M^cKernan-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(K_X+B) is Cartier and nef for some fixed positive integer m, is bounded modulo flops. This is a joint work with Chen Jiang.

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.

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.

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.

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.

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 X via line arrangements and abelian covers.

23.07.2021
(Friday!)
at 10am!