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

# “Lines on Smooth
Quartics”

By

**ALEXANDER DEGTYAREV**

**(BİLKENT UNIVERSITY)**

**Abstract: **In 1943, B. Segre proved that a smooth quartic
surface in the complex projective space 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, 2015), 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 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. This
relatively simple reduction has lead us to an
extremely difficult arithmetical problem. Nevertheless, the approach turned out
quite fruitful: for the moment, we have a complete classification of smooth quartics containing more than 52 lines. As an immediate
consequence of this classification, we have the following:

-- an
alternative proof of Segre's bound 64;

-- Shur's
quartic is the only one with 64 lines;

-- a
real quartic may contain at most 56 real lines;

-- a
real quartic with 56 real lines is also unique;

-- the
number of lines takes values {0,...,52,54,56,60,64}.

Conjecturally, we have a
complete list of all quartics with more than 48
lines; there are about two dozens of species, most projectively
rigid.

I will discuss methods used
in the proof and a few problems that are still open, e.g., the minimal fields
of definition, triangle-free configurations, lines in singular quartics, etc. This subject is a joint work in progress
with Ilia Itenberg and Sinan
Sertöz.

** **

**Date: ****Friday, October 02, 2015**

**Time: ****15.40**

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

Tea and cookies will be served before the
seminar.