**Department of Mathematics**

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

# “Lines in K3
surfaces II”

By

**ALEXANDER DEGTYAREV**

**(BİLKENT UNIVERSITY)**

**Abstract: **This is a continuation of last week's talk. The
unifying theme of this series of talks is the classical problem of counting
lines in the projective models of K3-surfaces of small degree. Starting with such classical results as Schur's
quartic and Segre's bound (proved by Rams and Schütt)
of 64 lines in a nonsingular quartic, I will discuss
briefly our recent contribution (with I. Itenberg and
A. S. Sertöz), i.e., the complete classification of nonsingular quartics with many
lines.

There are limitless opportunities in extending and generalizing these
results. First, one can switch from the complex numbers to an algebraically
closed field of characteristic p>0. Here, of course, most interesting are
the so-called (Shioda) supersingular
surfaces. I will discuss the properties of (quasi-)elliptic pencils on such
surfaces, culminating in the classification of large configurations of lines
for p=2,3. Alternatively, one may consider non-closed
fields such as R or Q. For the former, the sharp bound is 56 real lines in a
real quartic; for the latter, the current bound is 52, and the best known
example has 46 lines.

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**Date: ****Friday, October 14, 2016**

**Time: ****15.40-**

**Place: ****Mathematics Seminar Room,
ODTÜ**

You are most cordially invited Tea and cookies
will be served before the seminar.