[Turkmath:10042] ODTU-Bilkent Algebraic Geometry Seminar-Bilkent-335

[Ali Sinan Sertoz]

Welcome to the 2014 Fall talks of ODTU-Bilkent Algebraic Geometry Seminars.*

This term we will start with two talks summarizing the recent 
developments on "Lines on Surfaces".

After that we are starting a learning seminar on dessins d'enfants. We 
will mostly be reading the following book:
Girondo and Gonzalez-Diez, /Introduction to Compact Riemann Surfaces and 
Dessins d'Enfants/,
London Mathematical Society Student Texts 79, Cambridge University 
Press, 2012.
The chapters are already shared among the volunteers so you can come to 
listen with no strings attached! :-)
*
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*
This week the ODTU-Bilkent Algebraic Geometry Seminar 
<http://www.bilkent.edu.tr/%7Esertoz/agseminar.htm>  is at *Bilkent.*

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Speaker: Alexander Degtyarev*
*Affiliation: *Bilkent
*
Title: Lines on surfaces-I*
*Abstract: *This is a joint project with I. Itenberg and S. Sertöz. I 
will discuss the recent developments in our never ending saga on lines 
in nonsingular projective quartic surfaces. In 1943, B. Segre proved 
that such a surface cannot contain more than 64 lines. (The champion, 
so-called Schur's quartic, has been known since 1882.) Even though a gap 
was discovered in Segre's proof (Rams, Schütt), the claim is still 
correct; moreover, it holds over any field of characteristic other than 
2 or 3. (In characteristic 3, the right bound seems to be 112.) At the 
same time, it was conjectured by some people that not any number between 
0 and 64 can occur as the number of lines in a quartic. We tried to 
attack the problem using the theory of K3-surfaces and arithmetic of 
lattices. Alas, a relatively simple reduction has lead us to an 
extremely difficult arithmetical problem. Nevertheless, the approach 
turned out quite fruitful: for the moment, we can show that there are 
but three quartics with more than 56 lines, the number of lines being 64 
(Schur's quartic) or 60 (two others). Furthermore, we can prove that a 
real quartic cannot contain more than 56 real lines, and we have an 
example realizing this bound. We can also construct quartics with any 
number of lines in {0; : : : ; 52; 54; 56; 60; 64}, thus leaving only 
two values open. Conjecturally, we have a list of all quartics with more 
than 48 lines. (The threshold 48 is important in view of another theorem 
by Segre, concerning planar sections.) There are about two dozens of 
species, all but one 1-parameter family being projectively rigid.
*
Date: *26 September 2014, Friday*
Time: *15:40+
*Place: *MathematicsSeminar Room, *Bilkent.**


* Tea and cookies will be served before the talk.

You are most cordially invited.

Ali Sinan Sertöz

-- 
Bilkent University, Department of Mathematics, 06800 Ankara, Turkey
Office: (90)-(312) - 290 1490
Department: (90)-(312) - 266 4377
Mobile: (90) 532 767 9654
Fax: (90)-(312) - 290 1797
e-mail:sertoz at bilkent.edu.tr
Web:http://www.bilkent.edu.tr/~sertoz
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