ODTÜ-BİLKENT Algebraic
Geometry Seminar
(See all past talks ordered according
to speaker or date)
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**** 2025 Spring Talks ****
This
semester we plan to have all of our seminars online
Alexander
Degtyarev - [Bilkent] - Split hyperplane sections on polarized
K3-surfaces
| Abstract: I will discuss
a new result which is an unexpected outcome,
following a question by Igor Dolgachev, of a long
saga about smooth rational curves on
(quasi-)polarized $K3$-surfaces. The best known
example of a $K3$-surface is a quartic surface in
space. A simple dimension count shows that a
typical quartic contains no lines. Obviously, some
of them do and, according to B.~Segre, the maximal
number is $64$ (an example is to be worked out).
The key r\^ole in Segre's proof (as well as those
by other authors) is played by plane sections that
split completely into four lines, constituting the
dual adjacency graph $K(4)$. A similar, though
less used, phenomenon happens for sextic
$K3$-surfaces in~$\mathbb{P}^4$ (complete
intersections of a quadric and a cubic): a split
hyperplane section consists of six lines, three
from each of the two rulings, on a hyperboloid
(the section of the quadric), thus constituting a
$K(3,3)$. Going further, in degrees $8$ and $10$ one's sense of beauty suggests that the graphs should be the $1$-skeleton of a $3$-cube and Petersen graph, respectfully. However, further advances to higher degrees required a systematic study of such $3$-regular graphs and, to my great surprise, I discovered that typically there is more than one! Even for sextics one can also imagine the $3$-prism (occurring when the hyperboloid itself splits into two planes). The ultimate outcome of this work is the complete classification of the graphs that occur as split hyperplane sections (a few dozens) and that of the configurations of split sections within a single surface (manageable starting from degree $10$). In particular, answering Igor's original question, the maximal number of split sections on a quartic is $72$, whereas on a sextic in $\mathbb{P}^4$ it is $40$ or $76$, depending on the question asked. The ultimate champion is the Kummer surface of degree~$12$: it has $90$ split hyperplane sections. The tools used (probably, not to be mentioned) are a fusion of graph theory and number theory, sewn together by the geometric insight. |
John Christian Ottem - [Oslo] - Fano varieties with torsion in the third cohomology group
|
Abstract: I will
explain a construction of Fano varieties with
torsion in their third cohomology group. The
examples are constructed as double covers of
linear sections of rank loci of symmetric
matrices, and can be seen as higher-dimensional
analogues of the Artin– Mumford threefold. As an
application, we will answer a question of Voisin
on the coniveau and strong coniveau filtrations of
rationally connected varieties. |
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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.
Talks of previous
years
