ODTÜ-BİLKENT Algebraic
Geometry Seminar
(See all past talks ordered according
to speaker or date)
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**** 2023 Spring Talks ****
This
semester we plan to have our seminars online
Salvatore Floccari-[Hannover]
- Sixfolds of generalized Kummer type
and K3 surfaces
|
Abstract: The
classical Kummer construction associates a $K3$
surface to any 2-dimensional complex torus. In my
talk I will present an analogue of this
construction, which involves the two most
well-studied deformation types of hyper-Kähler
manifolds in dimension 6. Namely, starting from
any hyper-Kähler sixfold $K$ of generalized Kummer
type, I am able to construct geometrically a
hyper-Kähler manifold of $K3^{[3]}$-type. When $K$
is projective, the associated variety is
birational to a moduli space of sheaves on a
uniquely determined $K3$ surface. As application
of this construction I will show that the
Kuga-Satake correspondence is algebraic for many
$K3$ surfaces of Picard rank 16. |
Domenico Valloni-[Hannover]
- Rational points on the
Noether-Lefschetz locus of K3 moduli spaces
|
Abstract: Let L
be an even hyperbolic lattice and denote by
$\mathcal{F}_L$ the moduli space of L-polarized K3
surfaces. This parametrizes K3 surfaces $X$
together with a primitive embedding of lattices $L
\hookrightarrow \mathrm{NS}(X)$ and, when $L =
\langle 2d \rangle $, one recovers the classical
moduli spaces of 2d-polarized K3 surfaces. In this
talk, I will introduce a simple criterion to
decide whether a given $\overline{
\mathbb{Q}}$-point of $\mathcal{F}_L$ has
generic Néron-Severi lattice (that is,
$\mathrm{NS}(X) \cong L$). The criterion is of
arithmetic nature and only uses properties of
covering maps between Shimura varieties. |
|
Abstract: I will
discuss bounds on the number of rational curves of
fixed degree on surfaces of various types with
special emphasis on polarized Enriques surfaces.
In particular, I will sketch the proof of the
bound of at most 12 rational curves of degree at
most d on high-degree Enriques
surfaces (based mostly on joint work with Prof. M.
Schuett (Hannover)). |
|
Abstract: Riemann's
theta function becomes polynomial when the
underlying curve degenerates to a singular curve.
We will give a classification of such curves
accompanied by historical remarks on the topic. We
will touch on relations of such theta functions
with solutions of the Kadomtsev-Petviashvili
hierarchy if time permits. |
|
Abstract: In this talk I will present a
class of non-simply connected Calabi-Yau 3-folds
with positive Euler characteristic which are the
quotient spaces of fixed-point-free group actions
on desingularizations of singular Schoen 3-folds.
A Schoen 3-fold is the fiber product of two
rational elliptic surfaces with section. Smooth
Schoen 3-folds are simply connected CY 3-folds.
Desingularizations of certain singular Schoen
3-folds by small resolutions have the same
property. If a finite group G acts freely on such
a 3-fold, the quotient is again a CY 3-fold. I
will present how to classify such group actions
using the automorphism groups of rational elliptic
surfaces with section. The smooth Schoen 3-fold
case gives 0 Euler characteristic whereas the
singular case results in positive Euler
characteristic for the quotient CY threefolds. |
|
Abstract: In
this talk I will present a class of non-simply
connected Calabi-Yau 3-folds with positive Euler
characteristic which are the quotient spaces of
fixed-point-free group actions on
desingularizations of singular Schoen 3-folds. A
Schoen 3-fold is the fiber product of two rational
elliptic surfaces with section. Smooth Schoen
3-folds are simply connected CY 3-folds.
Desingularizations of certain singular Schoen
3-folds by small resolutions have the same
property. If a finite group G acts freely on such
a 3-fold, the quotient is again a CY 3-fold. I
will present how to classify such group actions
using the automorphism groups of rational elliptic
surfaces with section. The smooth Schoen 3-fold
case gives 0 Euler characteristic whereas the
singular case results in positive Euler
characteristic for the quotient CY threefolds. |
|
Abstract: An extremal
Kähler metric is a canonical Kähler metric,
introduced by E. Calabi, which is somewhat more
general than a constant scalar curvature Kähler
metric. The existence of such a metric is an
ongoing research subject and expected to be
equivalent to some form of geometric stability of
the underlying polarized complex manifold $(M, J,
[\omega])$ –the Yau-Tian-Donaldson conjecture.
Thus it is no surprise that there is a moment map,
the scalar curvature (A. Fujiki, S. Donaldson),
and the problem can be described as an infinite
dimensional version of the familiar finite
dimensional G.I.T. |
|
Abstract: Motivated by
applications to the theory of error-correcting
codes, we give an algorithmic method for computing
a generating set for the ideal generated by
$\beta$-graded polynomials vanishing on a subset
of a simplicial complete toric variety $X$ over a
finite field $\mathbb{F}_q$, parameterized by
rational functions, where $\beta$ is a $d\times r$
matrix whose columns generate a subsemigroup
$\mathbb{N}\beta$ of $\mathbb{N}^d$. We also give
a method for computing the vanishing ideal of the
set of $\mathbb{F}_q$-rational points of $X$. We
talk about some of its algebraic invariants
related to basic parameters of the corresponding
evaluation code. When $\beta=[w_1 \cdots w_r]$ is
a row matrix corresponding to a numerical
semigroup $\mathbb{N}\beta=\langle w_1,\dots,w_r
\rangle$, $X$ is a weighted projective space and
generators of its vanishing ideal is related to
the generators of the defining (toric) ideals of
some numerical semigroup rings corresponding to
semigroups generated by subsets of
$\{w_1,\dots,w_r\}$. |
|
Abstract: In this talk we will start with
basics of moduli space of curves, coverings of
curves, p-ranks and mention the differences in
characteristics 0 and positive
characteristics.Then we'll define Prym variety
which is a central object of study in arithmetic
geometry like Jacobian variety. The goal
of the talk is to understand various existence
results about Prym varieties of given genus,
p-rank and characteristics of the base field.
This is joint work with Rachel Pries. |
|
Abstract: Counting or estimating the
number of lines or, more generally, low degree
rational curves on a polarized algebraic surface
is a classical problem going back almost 1.5
centuries. After a brief historical excurse, I
will try to give an account of the considerable
progress made in the subject in the last decade
or so, mainly related to various
(quasi-)polarizations of K3-surfaces:
If time permits, I will briefly
discuss other surfaces/varieties as well. |
ODTÜ talks are either at Hüseyin Demir Seminar room or
at Gündüz İkeda seminar room at the
Mathematics building of ODTÜ.
Bilkent talks are at room 141 of Faculty of Science
A-building at Bilkent.
Zoom talks are online.
