**ODTÜ-B****İ****LKENT
ALGEBRAIC GEOMETRY SEM****I****NAR**

# “Lines on Surfaces-I”

By

**ALEXANDER DEGTYAREV**

**(BİLKENT UNIVERSITY)**

**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: ****Friday, September 26, 2014**

**Time: ****15.40**

**Place: ****Mathematics Seminar Room,
****SA – 141**

Tea and cookies will be served before the seminar.