Department of Mathematics
By
ALEXANDER DEGTYAREV
(BİLKENT UNIVERSITY)
Abstract: 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.
Date: Friday, October 7, 2016
Time: 15.40-
Place: Mathematics Seminar Room, SA – 141
You are most cordially invited Tea and cookies
will be served before the seminar.