computing l{functions of superelliptic curvesschuett/abstractag.pdf · fourfold y that does not...

DMV-PTM Mathematical Meeting17–20.09.2014, Poznań

Computing L–functions of superelliptic curves

Irene Bouw

Universität Ulm, [email protected]

The talk is based on the joint work with Stefan Wewers

Session: Algebraic Geometry

In this talk we discuss an approach for computing the L–functions of a

curve via stable reduction. We focus on superelliptic curves X defined over

a number field, which are given by an equation yn = f(x). We compute the

stable reduction of X at primes p whose residue characteristic is prime to n.

We then use this information to compute the local L–factor and the conductor

at p.


Geometry of moduli spaces of Higgs bundles oncurves.

Jochen Heinloth

Universität Duisburg-Essen, [email protected]

A part of this talk is based on joint work with O. Garćıa-Prada and A. Schmitt


Moduli spaces of Higgs bundles possess a wealth of unusual geometric prop-

erties that made them appear in very different contexts: They admit a natural

family of different complex structures, form an integrable system, give a geo-

metric description of terms appearing in the trace formula, are diffeomorphic to

character varieties, etc. In the talk I will try to explain some of these structures

and report on some recent results and open questions on the geometry of these

spaces.


A uniformization for the moduli space of abelianvarieties of dimension six

Gavril Farkas

Humboldt Universität zu Berlin, [email protected]

Session: Algebraic geometry

The general principally polarized abelian variety of dimension at most five is

known to be a Prym variety. This reduces the study of abelian varieties of small

dimension to the beautifully concrete theory of algebraic curves. I will discuss

recent progress on finding a structure theorem for principally polarized abelian

varieties of dimension six, and the implications this uniformization result has

on the geometry of the moduli space A6.


Calabi–Yau threefolds in P6

Micha l Kapustka

Jagiellonian University in Kraków, [email protected]

The talk is based on the joint work with Grzegorz Kapustka


Calabi–Yau threefolds in P6, so–called Pfaffian Calabi–Yau threefolds, area special class of Calabi–Yau threefolds which on one hand have often precise

descriptions in terms of equations and on the other are hard to study from the

point of view of mirror symmetry. In this talk, we shall review the theory of

these manifolds and present directions for possible future investigation.


Twisted cubics on cubic fourfolds

Manfred Lehn

Johannes Gutenberg–Universität, Mainz, [email protected]

This is a report on joint work with C. Lehn, C. Sorger, D. van Straten, andwith N. Addington


The moduli space of generalised twisted cubic curves on a smooth cubic

fourfold Y that does not contain a plane is shown to be smooth, 10-dimensional

and projective, and to admit a contraction to an 8-dimensional smooth variety

Z(Y ) that is irreducible holomorphic symplectic. Varying Z(Y ) with Y gives

a complete 20-dimensional family of projective holomorphic symplectic mani-

folds. If Y is a pfaffian cubic, Z(Y ) is birational to the fourth Hilbert scheme

of points on the K3-surface associated to Y by Beauville–Donagi.


Hurwitz spaces of torus covers:irreducibility conjectures and degree calculations

Martin Möller

Goethe Universität Frankfurt, [email protected]


Hurwitz spaces for covers of the projective line or with many branch points

are connected and their degree in known by representation theory. Here, on the

contrary, we consider Hurwitz spaces for branched covers of the torus branched

over one point only. Interest in this particular case stems from the theory of

Teichmüller curves.

Even for genus two covers, the components of these Hurwitz spaces are only

conjecturally known. We present these conjectures, compute the degree of the

Hurwitz spaces and their classes in the Picard groups of split Hilbert modular

surfaces. The method relies on theta functions and intersection theory on the

universal family of abelian surfaces.

This is joint work with André Kappes


A Gysin formula for Hall-Littlewood polynomials

Piotr Pragacz

Polish Academy of Scinces, [email protected]


Schubert calculus on Grassmannians is governed by Schur S-functions, the

one on Lagrangian Grassmannians by Schur Q-functions. There were several

attempts to give a unifying approach to both situations. We propose to use

Hall-Littlewood symmetric polynomials (invented by Ph. Hall in the 1950s in

his study of the combinatorial lattice structure of finite abelian p-groups). With

the projection in a Grassmann bundle, there is associated its Gysin map, in-

duced by pushing forward cycles (topologists call it ”integration along fibers”).

We state and prove a Gysin formula for HL-polynomials in these bundles. We

discuss its two specializations, giving better insights to previously known for-

mulas.


On Enriques surfaces with four cusps

S lawomir Rams

Jagiellonian University/Leibniz University Hannover, [email protected]

The talk is based on the joint work with M. Schütt (Hannover)


One can show that maximal number of A2–configurations on an Enriques

surface is four. In my talk I will classify all Enriques surfaces with four A2–

configurations. In particular I will show that they form two families in the

moduli of Enriques surfaces In particular, I will construct open Enriques sur-

faces with fundamental groups (Z/3Z) ⊕ (Z/2Z)⊕2 and Z/6Z, completing thepicture of the A2–case and answering a question put by Keum and Zhang.


Minkowski decomposition of Okounkov bodies

David Schmitz

Philipps-Universität Marburg, [email protected]


In recent years, the construction of Okounkov bodies for big line bundles on

normal projective varieties introduced by Lazarsfeld and Musţată and indepen-

dently by Kaveh and Khovanskii has raised quite a lot of interest. These convex

bodies carry important information on the sections of multiples of the line bun-

dle. Unfortunately, they are notoriously hard to determine. I report on results

of joint work with P. Luszcz-Świdecka and P. Pokora, S. Urbinati concerning a

new approach to describing these bodies as Minkowski sums of simple “builing

blocks”. I will also mention an application on the problem of polyhedrality of

global Okounkov bodies appearing in joint work with H. Seppänen.


Differential operators of Calabi-Yau type

Duco van Straten

Johannes Gutenberg–Universität, Mainz, [email protected]


In the talk I will report on the ongoing hunt for the differential operators

that have the properties abstracted from those coming from one parameter

families of Calabi-Yau threefolds and which by mirror symmetry are supposed

to be related to the Gromov-Witten invariants of Calabi-Yau threefolds with

Picard number one.


Configurations of lines with triple points.

Halszka Tutaj-Gasińska

Jagiellonian University, [email protected]

The talk is based on the joint work with M.Dumnicki, T.Szemberg, J.Szpond.


I will speak about some recent results involving some configurations of lines

with many triple points. In particular I will describe configurations giving

counterexamples to the I(3) ⊂ I2 containment problem, and I will discuss some(non)existence results.


Fano manifolds whose elementary contractionsare smooth P1-fibrations

Jaros law A. Wísniewski

Institute of Mathematics, University of Warsaw, [email protected]

The talk is based on the joint work with Gianluca Occhetta (Trento), Luis SolaConde (Madrid) and Kiwamu Watanabe (Saitama)


This presentation concerns a geometric characterization of complete flag

varieties for semisimple algebraic groups. Namely, if X is a Fano manifold

whose all elementary contractions are P1-fibrations then X is isomorphic to

the complete flag manifold G/B where G is a semi-simple Lie algebraic group

and B is a Borel subgroup of G.

Our proof of this statement is based on the following ideas: Every smooth

P1-fibration of X provides an involution of the vector space N1(X) of classes

of R-divisors in X. We show that these involutions generate a finite reflection

group, which is the Weyl group W of a semisimple Lie group G. Next we use

P1-fibrations of X to define a set of auxiliary manifolds called Bott-Samelson

varieties of X, which are analogues of the Bott-Samelson varieties that appear

classically in the study of Schubert cycles of flag varieties. Subsequently we

show that the recursive construction of appropriately chosen chain of Bott-

Samelson varieties depends only on the combinatorics of the Weyl group W

and ultimately we infer the isomorphism between X and G/B


On proper polynomial and holomorphic mappings

Zbigniew Jelonek

Polish Academy of Sciences, [email protected]


Let X,Y be smooth algebraic varieties of the same dimension. Let f, g :

X −→ Y be finite regular mappings. We say that f, g are equivalent if thereexists a regular automorphism Φ ∈ Aut(X) such that f = g ◦ Φ. Of course iff, g are equivalent, then they have the same discriminants (i.e., the same set of

critical values) and the same geometric degree. We show, that conversely there

is only a finite number of non-equivalent finite regular mappings f : X → Y,such that the discriminant D(f) = V and µ(f) = k. As one of applications we

show that if f : Cn → Cn is a proper mapping with D(f) = {x ∈ Cn : x1 = 0},then f is equivalent to the mapping g : Cn 3 (x1, ..., xn) 7→ (xk1 , x2, ..., xn) ∈Cn, where k = µ(f). Moreover, if f : X → Y is a finite mapping of topologicaldegree two, then there exists a regular automorphism Φ : X → X which actstransitively on fibers of f. In particular for n > 1 there is no finite mappings

f : Pn → Pn of topological degree two.We prove the same statement in the local (and sometimes global) holo-

morphic situation. In particular we show that if f : (Cn, 0) → (Cn, 0) is aproper and holomorphic mapping of topological degree two, then there exist

biholomorphisms Ψ,Φ : (Cn, 0)→ (Cn, 0) such that Ψ ◦ f ◦Φ(x1, x1, . . . , xn) =(x21, x2, . . . , xn). Moreover, for every proper holomorphic mapping f : (Cn, 0)→(Cn, 0) with smooth discriminant there exist biholomorphisms Ψ,Φ : (Cn, 0)→(Cn, 0) such that Ψ ◦ f ◦ Φ(x1, x1, . . . , xn) = (xk1 , x2, . . . , xn), where k = µ(f).


On the Abhyankar–Moh inequality

Arkadiusz P loski

Kielce University of Technology, [email protected]


Let C be a complex affine algebraic curve of degree n > 1 having only one

branch at infinity γ and let r0, r1, . . . , rh be the n–sequence of the semigroup

G of the branch γ defined as follows: r0 = n, rk = min{r ∈ G : r 6∈ Nro +· · · + Nrk−1} for k ≥ 1 and G = Nro + · · · + Nrh. Then the Abhyankar–Mohinequality (see [1, 2]) can be stated in the form

gcd{r0, . . . , rh−1}rh < n2. (AMn)

The aim of this talk is to present (see [3]) some results on the semigrups G ⊂ Nof plane branches γ with property (AMn). In particular we describe such

semigroups with the maximum conductor.

References

[1] S.S.Abhyankar, T.T.Moh, Embeddings of the line in the plane. J. reine angew.Math.276 (1975), 148-166.

[2] E.Garćıa Barroso, A.P loski, An approach to plane algebroid branches preprintarXiv:1208.0913 [math.AG].

[3] R.D.Barrolleta, E.R. Garca Barroso and A.P loski, Appendix to [2].

computing l{functions of superelliptic curvesschuett/abstractag.pdf · fourfold y that does not...

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