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<div class="moz-header-display-name" style="display:inline;">Subject:
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ODTU-Bilkent Algebraic Geometry Seminar-Zoom-560</td>
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<div class="moz-header-display-name" style="display:inline;">From:
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Ali Sinan Sertöz <a class="moz-txt-link-rfc2396E" href="mailto:sertoz@bilkent.edu.tr"><sertoz@bilkent.edu.tr></a></td>
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<div class="moz-header-display-name" style="display:inline;">Date:
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18/02/2025, 6:25 pm</td>
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<div align="center"><big><b><big>Welcome to the 2025
Spring talks of ODTU-Bilkent Algebraic Geometry
Seminars</big></b></big><b><br>
</b></div>
<div align="center"><i>since 2000</i><br>
</div>
<div align="center"><b><b>=================================================================</b></b><br>
<br>
This week the <a
href="http://www.bilkent.edu.tr/~sertoz/agseminar.htm" target="_blank">ODTU-Bilkent
Algebraic Geometry Seminar</a> is <b>online</b><br>
<br>
<i><font color="#ff00ff">This talk will begin at <u><b>15:40</b></u><u> (GMT+3)</u></font></i><br>
<a
href="https://www.timeanddate.com/worldclock/fixedtime.html?msg=ODT%C3%9C-Bilkent+Algebraic+Geometry+Seminar&iso=20250228T1540&p1=19&ah=1"
target="_blank">Please check your time difference
between Ankara and your city here</a><br>
<b>=================================================================</b></div>
<div align="center">
<div align="center"><img
src="cid:part1.anBI6Aze.9QyOCX80@bilkent.edu.tr"
alt="" class=""></div>
</div>
<div align="center"><br>
</div>
<div align="center"><i>Van Gogh (1853-1890)</i><br>
<div align="left"><b><b><font color="#ff0000">Speaker:
</font><a
href="http://www.fen.bilkent.edu.tr/%7Edegt/">Alexander
Degtyarev</a><font color="#ff0000"><font
color="#000000"> <br>
</font></font></b></b><b><b><font
color="#ff0000">Affiliation: </font><i>Bilkent</i></b></b></div>
<div align="left"><b><b><i><br>
</i></b></b></div>
<div align="left"><b><b><font color="#ff0000"> Title:
</font></b>Split hyperplane sections on
polarized K3-surfaces<br>
<b><font color="#ff0000"><font color="#000000"><br>
</font></font></b></b></div>
</div>
<div align="left"><b><b><font color="#ff0000">Abstract:
</font></b></b>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 $K(3,3)$.<br>
<br>
<div align="justify">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).<br>
</div>
<br>
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<br>
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.<br>
<br>
The tools used (probably, not to be mentioned) are a
fusion of graph theory and number theory, sewn
together by the geometric insight.<br>
<br>
</div>
<div align="left"><font color="#ff0000"><font
color="#000000"><br>
</font></font></div>
<div align="left"><font color="#ff0000"><font
color="#000000"><b><font color="#ff0000">Date:<font
color="#000000"> 28 February 2025</font></font></b>,
<b>Friday</b></font></font><br>
</div>
<div align="left"><font color="#ff0000"><font
color="#000000"><b><font color="#ff0000">Time: </font>15:40 <i>(GMT+3)</i></b><br>
<b><font color="#ff0000">Place: </font></b><font
color="#ff0000"><font color="#000000"><b>Zoom</b></font></font></font></font></div>
<blockquote>
<p><font color="#ff0000"><font color="#000000"><font
color="#ff0000"><font color="#000000"><b><b
style="color:rgb(51,51,51);font-family:arial,verdana,sans-serif;font-size:14px">One
day before the seminar, an announcement
with the Zoom meeting link will be sent
to those who registered with Sertöz.<br>
</b></b></font></font></font></font></p>
<p><font color="#ff0000"><font color="#000000"><font
color="#ff0000"><font color="#000000"><b><b
style="color:rgb(51,51,51);font-family:arial,verdana,sans-serif;font-size:14px">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.<br>
</b></b></font></font></font></font></p>
<p><font color="#ff0000"><font color="#000000"><font
color="#ff0000"><font color="#000000"><b><b
style="color:rgb(51,51,51);font-family:arial,verdana,sans-serif;font-size:14px">If
you have not registered before, please
contact him at <a
href="mailto:sertoz@bilkent.edu.tr?subject=Zoom%20Seminar%20Address%20Request"
target="_blank">sertoz@bilkent.edu.tr</a>.</b></b></font></font></font></font></p>
</blockquote>
<div align="left"><br>
</div>
<div align="left">You are most cordially invited to
attend.</div>
<div align="left"><br>
</div>
<div align="left">Ali Sinan Sertöz</div>
<div align="left"><font size="1"><i>(PS: <a
href="mailto:sertoz@gmail.com?cc=sertoz%40bilkent.edu.tr&subject=Unsubscribe&body=Please%20remove%20this%20address%20from%20the%20announcement%20list%20of%20ODT%C3%9C-Bilkent%20Algebraic%20Geometry%20Seminars.">To
unsubscribe from this list please click here and
send the custom mail without changing anything</a>.)</i></font>
</div>
<hr>
<pre cols="72">----------------------------------------------------------------------------
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: <a href="mailto:sertoz@bilkent.edu.tr" target="_blank"
class="gmail-moz-txt-link-freetext moz-txt-link-freetext">sertoz@bilkent.edu.tr</a>
Web: <a href="http://sertoz.bilkent.edu.tr" target="_blank">sertoz.bilkent.edu.tr</a>
----------------------------------------------------------------------------</pre>
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