IMPANGA is an algebraic geometry seminar organized by Piotr Achinger, Jarosław Buczyński,
and Michał Kapustka. In the academic year 2024/25, the seminar
meets twice per month for a one day session on Friday,
with two 60 min talks separated by a lunch break (11:00-12:00
and 13:30-14:30). IMPANGA meets in Room 403 at IMPAN
(unless stated otherwise).
To receive notifications about upcoming seminars join
impanga-mailing-list@impan.pl
using google account, or contact one of the organizers.
IMPANGA was founded at IMPAN in 2000 by late Piotr Pragacz. See here for information on former meetings of IMPANGA
Upcoming meeting
Jan 10 (impanga 463)
K-stablity of Fano threefold hypersurfaces of index 1
Speaker: Livia Campo
11:00–12:00, IMPAN 403
Abstract
The existence of Kaehler-Einstein metrics on Fano 3-folds
can be determined by studying lower bounds of stability
thresholds. An effective way to verify such bounds is to
construct flags of point-curve-surface inside the Fano
3-folds. This approach was initiated by Abban-Zhuang, and
allows us to restrict the computation of bounds for
stability thresholds only on flags. We employ this machinery
to prove K-stability of terminal quasi-smooth Fano 3-fold
hypersurfaces. This is deeply intertwined with the geometry
of the hypersurfaces: in fact, birational rigidity and
superrigidity play a crucial role. The superrigid case had
been attacked by Kim-Okada-Won. In this talk I will discuss
the K-stability of strictly rigid Fano hypersurfaces via
Abban-Zhuang Theory. This is a joint work with Takuzo Okada.
A new technique for lower bounding the border rank
Speaker: Tomasz Mańdziuk
13:30–14:30, IMPAN 403
Abstract
A
fundamental problem in the theory of tensors is establishing
lower bounds for the border ranks of explicit tensors. In
the talk I will present a technique for establishing lower
bounds
on border rank based on border apolarity and I will discuss
progress concerning the border
rank of the 3x3 matrix multiplication tensor. The talk is
based on a joint work with Amy Huang,
Austin Conner and J.M. Landsberg.
Future meetings
Jan 24 (impanga 464)
TBA
Speaker: TBA
11:00–12:00, IMPAN 403
Abstract
tba
TBA
Speaker: TBA
13:30–14:30, IMPAN 403
Abstract
tba
Past meetings
Oct 18 (impanga 458)
Series of lectures on the Zariski multiplicity conjecture part I
A short introduction to the Zariski multiplicity
conjecture
Speaker: Christophe Eyral
(IMPAN)
11:00–12:00, IMPAN 403
Abstract
Stated in 1971, the Zariski multiplicity conjecture asserts
that the multiplicity of a reduced complex hypersurface
singularity is an embedded topological invariant. More than
50 years later, this problem is still open in general.
Several partial answers have been given over the years, with
a major breakthrough by J. Fernández de Bobadilla and T.
Pełka who recently proved a version of the conjecture for
1-parameter families of isolated singularities. In this
talk, I will give an overview of the most significant
results that have been obtained.
Symplectic monodromy at radius zero
Speaker: Tomasz Pełka (MIMUW)
13:30–14:30, IMPAN 403
Abstract
This is the first of two lectures concerning my joint work
with J. F. de Bobadilla, proving Zariski multiplicity
conjecture for μ-constant families of isolated hypersurface
singularities. In this lecture I will explain the key
ingredient of the proof, namely the construction of a
representative of monodromy which is an exact
symplectomorphism with very good dynamic properties. Given a
family of Kähler manifolds over a punctured disk, together
with a model with snc central fiber, I will show how to
extend it to a symplectic fibration over an annulus, such
that over the inner circle ("at radius zero") most choices
become irrelevant, in particular the symplectic monodromy
can be read off from the dual complex of the central fiber.
The resulting fibration, called the A'Campo space, can be
applied to many degeneration problems in algebraic geometry.
If time permits, I will show how to use it to produce
Lagrangian tori approximating those predicted by Mirror
Symmetry.
Oct 25 (impanga 459)
Series of lectures on the Zariski multiplicity conjecture part
II
Heegaard Floer homologies for algebraic geometer
Speaker: Maciej Borodzik (IMPAN)
11:00–12:00, IMPAN 403
Abstract
In
this introductory lecture, I will describe basics of
Heegaard Floer homology (construction and properties)
focusing on examples of special importance to experts in
singularity theory and low-dimensional algebraic geometry.
Equimultiplicity of μ-constant families
Speaker: Tomasz Pełka (MIMUW)
13:30–14:30, IMPAN 403
Abstract
In
the second lecture, I will explain how to use the A'Campo
space, to prove the Zariski conjecture. Just like the
classical topological A'Campo construction allows to compute
Lefschetz number of monodromy's iterates, our symplectic
version yields a spectral sequence converging to its Floer
homology. This proof is a slight generalization of the
previous work by McLean, in particular it recovers his
characterization of multiplicity as the smallest integer m
such that the m-th iterate of monodromy has nonvanishing
Floer homology. Our more general setting allows us to apply
it to each member of a μ-constant family, and infer that the
resulting Floer homology - hence multiplicity - stays the
same.
Nov 15 (impanga 460)
Series of lectures on the Zariski multiplicity conjecture part
III
Bi-Lipschitz equivalent cones with different degrees
Speaker: Zbigniew Jelonek
(IMPAN)
11:00–12:00, IMPAN 403
Abstract
We show that for every $k\ge 3$ there exist complex
algebraic cones of dimension $k$ with isolated
singularities, which are bi-Lipschitz and
semi-algebraically equivalent but have different degrees.
We also prove that homeomorphic projective hypersurfaces
with dimension greater than 2 have the same degree. In the
final part of the paper, we classify links of real cones
with base $\mathbb{P}^1\times \mathbb{P}^2.$ As an
application we give an example of three four-dimensional
real algebraic cones in $\mathbb{R}^8$ with isolated
singularity which are semi-algebraically and bi-Lipschitz
equivalent but have non-homeomorphic bases. We discover
also some new tools to study the links of real algebraic
varieties. Moreover, we give examples of real manifolds,
which are not diffeomorphic to projective manifolds of odd
degree.
A'Campo space: construction and applications.
Speaker: Tomasz Pełka (MIMUW)
13:30–14:30, IMPAN 403
Abstract
I will explain in detail the construction of the A'Campo
space. It plays a crucial role in the proof of Zariski
Multiplicity Conjecture, as it provides a representative
of monodromy with good dynamics. In this lecture, I will
focus on other applications of this construction. For
example, in the context of maximal Calabi-Yau
degenerations, it provides a family of Kahler metrics with
properties similar to those expected from the Ricci flat
ones. The key idea is to perform most computations at the
boundary of the A'Campo space, and then push the result
back to the original degeneration using the symplectic
connection.
Nov 22 (impanga 461)
Degree of the subspace variety
Speaker: Pierpaola
Santarsiero (University of Bologna)
11:00–12:00, IMPAN 403
Abstract
Subspace varieties are algebraic varieties whose
elements are tensors with bounded multilinear rank. In
this talk, we understand how to compute their degrees by
computing their volumes. This is joint work with P.
Breiding.
Prym maps of non-cyclic coverings
Speaker: Paweł Borówka (UJ)
13:30–14:30, IMPAN 403
Abstract
Prym theory connects coverings of curves with abelian
varieties. The most famous result of the theory, due to
Donagi and Smith, is the fact that the Prym map of
double coverings of genus 6 curves is 27:1 and is
related to 27 lines on a cubic. Apart from that, we know
a lot about Prym maps of double coverings and we have
some results for cyclic coverings. After a brief
introduction to the theory and mentioned results, I will
focus on my results about Klein (i.e. Z_2xZ_2) coverings
of hyperelliptic curves. The talk is based on joint
projects with A. Ortega and with A. Shatsila.
Dec 06 (impanga 462)
Free plane curves in algebraic geometry
Speaker: Piotr Pokora
(UKEN Kraków)
11:00–12:00, Kraków branch of
IMPAN, św. Tomasza 30/7 Kraków
Abstract
I will present the recent developments on free
plane curves, mostly focusing on the problem of
constructing algebraic surfaces having large Picard
numbers and the so-called Numerical Terao's
Conjecture. Time permitting, I will deliver some
recent progress on Ziegler pairs of line arrangements.
Cactus scheme, catalecticant minors and
singularities of secant varieties to high degree
Veronese reembeddings.
Speaker: Jarosław
Buczyński (IMPAN)
13:30–14:30, Kraków branch of
IMPAN, św. Tomasza 30/7 Kraków
Abstract
The r-th cactus variety of a subvariety X in a
projective space generalises secant variety of X and
it is defined using linear spans of finite schemes of
degree r. It's original purpose was to study the
vanishing sets of catalecticant minors. We propose
adding a scheme structure to the cactus variety and we
define it via relative linear spans of families of
finite schemes over a potentially non-reduced base. In
this way we are able to study the vanishing scheme of
the catalecticant minors. For X which is a
sufficiently large Veronese reembedding of projective
variety, we show that r-th cactus scheme and the zero
scheme of appropriate catalecticant minors agree on an
open and dense subset which is the complement of the
(r-1)-st cactus variety/scheme. As an application, we
can describe the singular locus of (in particular)
secant varieties to high degree Veronese varieties.
Based on a joint work with Hanieh Keneshlou.