[Turkmath:6313] ODTU-Bilkent Algebraic Geometry Seminar-Zoom-540

[Ali Sinan Sertöz]

*Welcome to the 2023 Fall talks of ODTU-Bilkent Algebraic Geometry 
Seminars**
*
/since 2000/
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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=20231215T1540&p1=19&ah=1>
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*
*/Johannes Vermeer (1632-1675)
Girl with a pearl earring (1665)/
**Speaker: Alexander Degtyarev <http://www.fen.bilkent.edu.tr/%7Edegt/>
****Affiliation: /Bilkent/****
Title:Lines on singular quartic surfaces via Vinberg

****
**
**Abstract: Large configurations of lines (or, more generally, rational 
curves of low degree) on algebraic surfaces  appear in various contexts, 
but only in the case of cubic surfaces the picture is complete. Our 
principal goal is the classification of large configurations of lines on 
quasi-polarized K3-surfaces /in the presence of singularities/. To the 
best of our knowledge, no attempt has been made to attack this problem 
from the lattice-theoretical, based on the global Torelli theorem, point 
of view; some partial results were obtained  by various authors using 
``classical'' algebraic geometry, but very little is known. The 
difficulty is that, given a polarized N\'eron--Severi lattice, computing 
the classes of smooth rational curves depends on the choice of a Weyl 
chamber of a certain root lattice, which is not unique.

We show that this ambiguity disappears and the algorithm becomes 
deterministic provided that /sufficiently many classes of lines are 
fixed/. Based on this fact, Vinberg's algorithm, and a combinatorial 
version of elliptic pencils, we develop an algorithm that, in principle, 
would list all extended Fano graphs. After testing it on octic 
K3-surfaces, we turn to the most classical case of simple quartics 
where, prior to our work, only an upper bound of 64 lines (Veniani, same 
as in the smooth case) and an example of 52 lines (the speaker) were 
known. We show that, /in the presence of singularities/, the sharp upper 
bound is indeed 52, substantiating the long standing conjecture (by the 
speaker) that the upper bound is reduced by the presence of smooth 
rational curves of lower degree.

We also extend the classification (I. Itenberg, A.S. Sertöz, and the 
speaker) of large configurations of lines on /smooth/ quartics down to 
49 lines. Remarkably, most of these configurations were known before.

This project was conceived and partially completed during our joint stay 
at the Max-Planck-Institut f\ür Mathematik, Bonn. The speaker is 
partially supported by TÜBİTAK project 123F111.
**

*Date:15 December 2023*, *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 send me a note.)/
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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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