[Turkmath:7029] ODTU-Bilkent Algebraic Geometry Seminar-Zoom-560

[Ali Sinan Sertöz]

Subject:
ODTU-Bilkent Algebraic Geometry Seminar-Zoom-560
From:
Ali Sinan Sertöz <sertoz at bilkent.edu.tr>
Date:
18/02/2025, 6:25 pm



*Welcome to the 2025 Spring talks of ODTU-Bilkent Algebraic Geometry 
Seminars**
*
/since 2000/
**=================================================================**

This week the ODTU-Bilkent Algebraic Geometry Seminar 
<http://www.bilkent.edu.tr/~sertoz/agseminar.htm>  is *online*

/This talk will begin at _*15:40*__ (GMT+3)_/
Please check your time difference between Ankara and your city here 
<https://www.timeanddate.com/worldclock/fixedtime.html?msg=ODT%C3%9C-Bilkent+Algebraic+Geometry+Seminar&iso=20250228T1540&p1=19&ah=1>
*=================================================================*

/Van Gogh (1853-1890)/
**Speaker: Alexander Degtyarev <http://www.fen.bilkent.edu.tr/%7Edegt/>
****Affiliation: /Bilkent/**
**/
/**
**Title: *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.


*Date:28 February 2025*, *Friday*
*Time: 15:40 /(GMT+3)/*
*Place: **Zoom*

    **One day before the seminar, an announcement with the Zoom meeting
    link will be sent to those who registered with Sertöz.
    **

    **If you have registered before for one of the previous talks, there
    is no need to register again; you will automatically receive a link
    for this talk too.
    **

    **If you have not registered before, please contact him at
    sertoz at bilkent.edu.tr
    <mailto:sertoz at bilkent.edu.tr?subject=Zoom%20Seminar%20Address%20Request>.**


You are most cordially invited to attend.

Ali Sinan Sertöz
/(PS: To unsubscribe from this list please click here and send the 
custom mail without changing anything 
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Ali Sinan Sertöz
Bilkent University, Department of Mathematics, 06800 Ankara, Türkiye
Office: (90)-(312) - 290 1490
Department: (90)-(312) - 266 4377
Fax: (90)-(312) - 290 1797
e-mail:sertoz at bilkent.edu.tr <mailto:sertoz at bilkent.edu.tr> 
Web:sertoz.bilkent.edu.tr <http://sertoz.bilkent.edu.tr> 
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