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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.
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DMV-PTM Mathematical Meeting17–20.09.2014, Poznań
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
Session: Algebraic Geometry
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.
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DMV-PTM Mathematical Meeting17–20.09.2014, Poznań
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.
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DMV-PTM Mathematical Meeting17–20.09.2014, Poznań
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
Session: Algebraic Geometry
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.
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DMV-PTM Mathematical Meeting17–20.09.2014, Poznań
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
Session: Algebraic Geometry
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.
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DMV-PTM Mathematical Meeting17–20.09.2014, Poznań
Hurwitz spaces of torus covers:irreducibility conjectures and degree calculations
Martin Möller
Goethe Universität Frankfurt, [email protected]
Session: Algebraic Geometry
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
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DMV-PTM Mathematical Meeting17–20.09.2014, Poznań
A Gysin formula for Hall-Littlewood polynomials
Piotr Pragacz
Polish Academy of Scinces, [email protected]
Session: Algebraic Geometry
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.
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DMV-PTM Mathematical Meeting17–20.09.2014, Poznań
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)
Session: Algebraic Geometry
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.
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DMV-PTM Mathematical Meeting17–20.09.2014, Poznań
Minkowski decomposition of Okounkov bodies
David Schmitz
Philipps-Universität Marburg, [email protected]
Session: Algebraic Geometry
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.
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DMV-PTM Mathematical Meeting17–20.09.2014, Poznań
Differential operators of Calabi-Yau type
Duco van Straten
Johannes Gutenberg–Universität, Mainz, [email protected]
Session: Algebraic Geometry
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.
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DMV-PTM Mathematical Meeting17–20.09.2014, Poznań
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.
Session: Algebraic Geometry
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.
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DMV-PTM Mathematical Meeting17–20.09.2014, Poznań
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)
Session: Algebraic Geometry
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
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DMV-PTM Mathematical Meeting17–20.09.2014, Poznań
On proper polynomial and holomorphic mappings
Zbigniew Jelonek
Polish Academy of Sciences, [email protected]
Session: Algebraic Geometry
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).
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DMV-PTM Mathematical Meeting17–20.09.2014, Poznań
On the Abhyankar–Moh inequality
Arkadiusz P loski
Kielce University of Technology, [email protected]
Session: Algebraic Geometry
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].