ODTÜBİLKENT
Algebraic Geometry Seminar
(See all past talks
ordered according to speaker
and date)
**** 2016 Fall Talks ****

Alexander Degtyarev[Bilkent]
 Lines in K3 surfaces
Abstract: The unifying
theme of this series of talks is the classical
problem of counting lines in the projective models
of $K3$surfaces of small degree. Starting with
such classical results as Schur's quartic and
Segre's bound (proved by Rams and Schütt) of $64$
lines in a nonsingular quartic, I will discuss
briefly our recent contribution (with I. Itenberg
and A. S. Sertöz), i.e., the complete
classification of nonsingular quartics with many
lines. Most quartics found (in an
implicit way) in our work are ``new'', attracting
the attention of experts in the field (Rams,
Schütt, Shimada, Shioda, Veniani). For example,
one of them turned out an alternative nonsingular
quartic model of the famous Fermat surface
$\Phi_4:=\{z_0^4+z_1^4+z_2^4+z_3^4=0\},$ raising
the natural question if there are other such
models. An extensive search (Shimada, Shioda)
returned no results, and we show that, although
there are over a thousand singular models,
only two models are smooth! Taking this
line of research slightly further, one can
classify all smooth quartic models of singular
$K3$surfaces of small discriminant, arriving at a
remarkable alternative characterisation of Schur's
quarticthe champion carrying $64$ lines: it is
also the (only) smooth quartic of the smallest
possible discriminant, which is $48$. Going even
further, we can study other projective models of
small degree; counting lines in these models, we
arrive at the following conjectures:
These conjectures are still wide
open; I only have but a few examples. 
Alexander Degtyarev[Bilkent]
 Lines in K3
surfacesII
Abstract:
This is the continuation of last week's
talk. 
Ali Sinan Sertöz[Bilkent]
 Introduction to complex K3 surfaces
Abstract: We will start
reviewing and explaining as the case might be some
introductory concepts in K3 surface theory. The
level will be introductory so it is a good
opportunity so jump on the "wagon". 
Ali Sinan Sertöz[Bilkent]
 K3 surfaces and lattices
Abstract:We will introduce some basic concepts of lattice theory that are used to understand K3 surfaces with a view towards Torelli type theorems. 
Ali Sinan Sertöz[Bilkent] K3 lattice of a K3 surface
Abstract: We will
continue our series on K3 surfaces by examining
the cohomology of K3 surfaces and finding out how
this cohomology structure characterizes the
surface. 
Abstract: In this talk
we discuss the problem of classifying complex
nonspecial simple quartics up to equisingular
deformation by reducing the problem to an
arithmetical problem about lattices. On this
arithmetical side, after applying Nikulin's
existence theorem, our computation based on the
MirandaMorrison's theory computing the genus
groups. We give a complete description of
equisingular strata of nonspecial simple
quartics. Finally we give ideas of the proof
of our principal result. 
Çisem Güneş[Bilkent]
 Classification of simple quartics up
to equisingular deformationII
Abstract: This is the
continuation of last week's talk. 
Oğuzhan Yörük[Bilkent]
 Which K3 surfaces of Picard rank 19
cover an Enriques surface?
Abstract: The
parities of the entries of the transcendental
lattice of a K3 surface $X$ determine, in most cases,
if $X$ covers an Enriques surface or not. We will
summarize what is known about this problem and talk
about the missing case when $\rho=19$. 
Alexander Degtyarev[Bilkent]
 Projective models of $K3$surfaces
Abstract: Now, that we
know everything about abstract $K3$surfaces, I
will try to take a closer look at SaintDonat's
seminal paper$^\ast$ and share my findings. This 
Ali Ulaş Özgür Kişisel  [ODTÜ]
 Arithmetic of K3 surfaces
Abstract:
I'll try to outline some of the
results in the survey paper of M. Schütt with the
same title. 
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
Abuilding at Bilkent.
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