Oberseminar

# Oberseminar

Oberseminar Algebraische und Arithmetische Geometrie

Das Oberseminar findet meist donnerstags von 16:30 Uhr bis 18 Uhr im Raum g117 oder f309 (*) statt.

Außerdem lassen sich die Oberseminartermine vergangener Semester abfragen.

WiSe 2013/14

Datum

Vortragender

Titel

10. Oktober (10:00-12:00)

Roberto Laface

On Zariski Decomposition With and Without Support

Abstract
Zariski Decomposition was firstly introduced by O. Zariski in 1962. In its original version, it allows to write any effective divisor D on a projective surface as the sum D=P+N of a nef divisor P, and an effective divisor N whose intersection matrix defines a negative definite quadratic form (they are respectively called positive and negative part of D). Later in 1979, the result was improved by T. Fujita, who extended it to pseudo-effective divisors. In 2008, Y. Miyaoka introduced the Zariski Decomposition for effective divisors with support in a negative definite cycle, which he needed in his study of the surfaces of general type: this version allows us to choose the negative part N.

The lecture aims at connecting the works of the authors presented in this timeline. After recalling the statements of Zariski and Fujita, we discuss the Zariski Decomposition with support, of which we give a different proof (following an idea of T. Bauer). Afterwards, we generalize Miyaoka's result in two ways: on one hand, we extend the result to any divisor D (so it does not have to be effective); on the other, we prove the result for pseudo-effective divisors with the relaxed hypothesis that the support is in any cycle, not necessarily negative definite. Finally, it comes naturally that the Zariski Decomposition for pseudo-effective divisors introduced by Fujita can be realized by iterating the Zariski Decomposition with support.

N.B.: The talk will be based on the following article:

http://arxiv.org/abs/1306.4697

17. Oktober

Alexey Basalev

3-dimensional Frobenius manifolds via elliptic curves

Abstract We investigate 3-dimensional Frobenius manifolds that appear as the subvarieties in the deformation space of simple elliptic singularities. We set up the connection between such Frobenius manifolds and elliptic curves. With the help of elliptic curves theory we give the classification of such Frobenius manifolds having additional symmetries and defined over Q.

24. Oktober

Slawek Rams

On quartic surfaces with many lines.

Abstract We show that a smooth quartic surface over algebraically closed field of characteristic different from 2,3 contains at most 64 lines. We discuss properties of quartics with lines of the second kind (joint work with Prof. M. Schuett (LU Hannover).

31. Oktover

Olivia Dumitrescu

Interpolation problems in P^n

Abstract We give an overview of interpolation problems in P^n. We investigate the speciality of effective linear systems in P^n with assigned multiple points and linear base locus. For this, we compute the cohomology groups of the strict transforms of divisors via the blow-up of their linear base locus. (with C. Brambilla and E. Postinghel) We extend this computations to non-effective divisors in P^n. In particular this shows that linear cycles are special effect varieties for effective linear systems interpolating points. Using the birational geometry of Cremona transformantion we give new classes of special effect varieties of dimension 1 and 2 in P^4. (with R. Miranda) This generalizes the elemetary (-1) curves of Laface-Ugalia in P^3, and it also give information on the Effective cone of the Blown up P^4 in arbitrary number of points.

14. November (16:00-17:00)

Elisabeth Werner

Algebraic geometric Codes and Lattices

Abstract We will give a short overwiew of codes and lattices and their connections. Then we want to construct codes and lattices from function fields of algebraic curves over finite fields. We fix a finite set of points. The code constructions are on the one hand evaluations of a linear system related to a divisor and on the other hand residues of linear systems of differentials. The lattice construction is related to lattices of units from number fields and makes use of principal divisors. In the end we will find a connection between this algebraic-geometric codes and lattices.

21. November

Discrete derived categories

Abstract Discrete derived categories, as defined by Vossieck, form a class of triangulated categories which are sufficiently simple to make explicit computation possible, but also non-trivial enough to manifest interesting behaviour. In this seminar, I will explain what they are, and talk about some recent work with D. Pauksztello and D. Ploog, where we study the autoequivalences and bounded t-structures on these categories. I will go on to describe work in progress, aiming to understand the corresponding spaces of Bridgeland stability conditions.

28. November

Christian Böhning (Hamburg)

Rationality problems: a bird's eye view

Abstract After a general introduction to and sample of guiding open
problems in the birational geometry of varieties close to the
rational ones, we discuss
(1) recent results for linear group quotients;
(2) the derived category and Hodge-theoretic approaches to
the irrationality problem for very general cubic fourfolds; their
shortcomings and ramifi cations (phantom categories); and
possible alternatives (dynamical spectra).

5. Dezember

Miguel Angel Marco Buzunariz

Euler characteristic, counting polynomial and Chern-Schwarz-MacPherson class of an algebraic set

Abstract Deducing the topology of an algebraic set from its defining equations is, in general, a difficult task. In this talk, a method to compute the Euler characteristic will be presented. Using the same method it is possible to compute a stronger invariant, that is directly related to the Chern-Schwarz-MacPherson class. We will also show how this stronger invariant is related to the problem of counting points of varieties over finite fields.

12. Dezember

Daniel Greb (Bochum)

Construction and variation of moduli spaces of sheaves on higher-dimensional base manifolds

Abstract While the variation of moduli spaces of H-slope/Gieseker-semistable sheaves on surfaces under change of the ample polarisation H is well-understood, research on the corresponding question in the case of higher-dimensional base manifolds revealed a number of pathologies. After presenting these, I will discuss recent joint work with Matei Toma (Nancy) and Julius Ross (Cambridge) which resolves some of these pathologies by looking at curves instead of divisors, and by embedding the moduli problem for sheaves into a moduli problem for quiver representations.

18. Dezember (10:00-12:00, g005)

Atsushi Takahashi

Weyl Groups and Artin Groups Associated to Orbifold Projective Lines

Abstract After explaining our motivation coming from mirror symmetry, we report on our recent study of a correspondence among orbifold projective lines, cusp singularities and cuspidal root systems. A conjectual relation among Weyl groups, Artin groups and the spaces of Bridgeland's stability conditions for some triangulated categories for orbifold projective lines will also be explained.

19. Dezember

Frederik Tietz

On homological stability of certain moduli spaces

Abstract (Co-)Homological stability phenomena for sequences of spaces or groups arise in many classical and modern contexts. In this talk, a few stability results are covered, including braid groups (Arnold), moduli of curves/mapping class groups (Harer, Madsen-Weiss), and Hurwitz spaces (Ellenberg-Venkatesh-Westerland). Emphasis lies on the presentation of the algebro-geometric and topological ideas that lie behind the theorems and their proofs. We will show how these results could fit into a common framework in order to obtain further stability results.

9. Januar

Matthias Zach

On the topology of simple isolated Cohen-Macaulay codimension 2 singularities in IC^5

Abstract The simple isolated Cohen-Macaulay codimension 2 singularities were classified by Anne Fruehbis-Krueger and Alexander Neumer. They have a smooth base in their semi-universal deformation and hence a unique Milnor fiber. Contrary to the case of hypersurface singularities, for Cohen-Macaulay codimension 2 singularities in IC^5 there can be two nonvanishing betti numbers contributing to the defect of the euler characteristic in a smoothing. Recently James Damon and Brian Pike did computations of the latter for some classified members revealing some peculiarities. I will present explicit computations of the vanishing cycles and their interplay for certain families and hope to discuss possible conjectures for the general behaviour.

16. Januar

Gunther Cornelissen (Utrecht)

Gonality of curves and graphs

Abstract The gonality of a smooth projective curve X over a field k is defined as the minimal degree of a non-constant morphism from X to the projective line. If k is the complex numbers, X can be considered as a compact Riemann surface, and Li and Yau have established a lower bound on the gonality of X in terms of the hyperbolic volume and the first eigenvalue of the Laplacian of X. Such a bound has numerous applications, for example, it leads to a lower bound on the gonality of modular curves for congruence groups that is linear in the index of the group in the full modular group (Abramovich) and to bounds on rational points in families of field extensions (Frey et. al.) I will present a non-archimedean analogue of these facts. The first result is a comparison theorem between gonality of a curve and the so-called stable gonality of its reduction graphs (a new concept). The main result is a lower bound for stable graph gonality in terms of the first eigenvalue of the Laplacian and some “volume” of the original graph (this can be seen as a substitute for graphs of the Li–Yau inequality from differential geometry, although the strict analogue of the original inequality fails for general graphs). We apply the results to give a lower bound for the gonality of arbitrary Drinfeld modular curves over finite fields and for general congruence subgroups Γ of analogous to the result of Abramovich. Another application is to rational points, and to lower bounds on the modular degree of certain elliptic curves over function fields (solving a problem of Papikian).

(joint work with Janne Kool and Fumiharu Kato)

20. Januar (11:00-12:00, f442)

Benjamin Bakker

The geometric Frey-Mazur conjecture

Abstract A crucial step in the proof of Fermat's last theorem was Frey's insight that a nontrivial solution would yield an elliptic curve with modular p-torsion but which was itself not modular. The connection between an elliptic curve and its p-torsion is very deep: a conjecture of Frey and Mazur, stating that the p-torsion group scheme actually determines the elliptic curve up to isogeny (at least when p>13), implies an asymptotic generalization of Fermat's last theorem. We study geometric analogs of this conjecture, and show that over function fields the map from isogeny classes of elliptic curves to their p-torsion group scheme is one-to-one. Our proof involves understanding curves on certain Shimura varieties, and fundamentally uses the interaction between its hyperbolic and algebraic properties. This is joint work with Jacob Tsimerman.

23. Januar

Malek Joumaah

Moduli spaces of K3^[2]-type fourfolds with non-symplectic involutions

Abstract For a K3 surface S and a non-symplectic involution i on S, Nikulin has shown that the deformation type of (S,i) is determined by the invariant sublattice of H^2(S,Z). The corresponding moduli space is an arithmetic quotient of a bounded symmetric domain, and in particular a quasi-projective variety. We will see that the latter statement can be generalized to K3^[2]-type fourfolds in a slightly weaker form. However, for a given invariant sublattice, there can be several deformation classes, and we will discuss a way to describe them.

30. Januar

Benjamin Wieneck

Discriminant loci of Lagrangian fibrations

Abstract Lagrangian fibrations on irreducible holomorphic symplectic manifolds are higher dimensional generalizations of elliptic K3 surfaces. The discrimiant locus of such a fibration i.e. the set which parametrizes the singular fibers is projective of codimension 1. A natural question is if there is a more natural scheme structure on the discriminant locus than the trivial reduced one. Such a structure could lead to a classification. I will discuss some concepts with a view to fibrations on K3^[n] which are induced by elliptic K3 surfaces.

4. Februar (14:15-15:45)

Thomas Dedieu

Limits of pluri--tangent planes to K3 surfaces

Abstract I will discuss the following general question : given a family $X \to D$ of complex surfaces in $\mathbf{P} ^N$ with smooth general member (here $D$ stands for the complex unit disk), what are the limits as $t \in D$ tends to $0$ of the families of hyperplanes tangent in $d$ points to $X _t$, $t \neq 0$ ? (here $d$ is a given integer, smaller than $N$) I will give a complete answer in the two cases of smooth quartic hypersufaces in $\mathbf{P} ^3$ degenerating to (i) the union of four planes, and (ii) a Kummer surface. This will then be applied to the study of the irreducibility and enumerative geometry of the Severi varieties of quartic $K3$ surfaces. Eventually, I will sketch a conjectural to the question in general. This is joint work with Ciro Ciliberto.

25. Februar (10-12)

Toshiyuki Katsura (Hosei, Tokyo)

Some invariants of algebraic varieties in positive characteristic

Abstract We discuss relations between certain invariants of varieties in positive characteristic, like the $a$-number and the height of the Artin-Mazur formal group. We calculate the $a$-number for Fermat surfaces. (joint-work with G. van der Geer)

27. Februar (14:15-15:45*)

Mathieu Dutour

Lattices and perfect forms

Abstract Lattices are discrete subgroups of R^n and they have interest in geometry of numbers, algebra (root lattices), number theory (mass formulas), and algebraic geometry. I will explain the basics of lattice theory, the classical examples and how to compute with them. Then I will explain the perfect form theory, which is the simplest example of a reduction theory and gives a tessellation of the space of positive definite quadratic forms. Finally, if time allows, I will try to give some other reduction theories (i.e. tessellation of the space of quadratic forms): The L-type reduction theory and the central cone compactification. Within the limits of my knowledge, I will present the relation between such tessellations and the corresponding toroidal compactification of the moduli space Ag of principally polarized abelian varieties. Again if time allows, I will present the possible applications of the tesselations to cohomology computations. (slides)

# Stellenausschreibung

Das Institut bietet zum 1. Januar 2014 eine Postdoc-Stelle an.

Das Institut bietet zum 1. April 2014 eine Postdoc-Stelle mit Spezialisierung auf Modulräume an.

# Workshop

A Superficial Afternoon am 6.12.2013 (Nikolaus) an der Universität Bielefeld.