ODTÜ-BİLKENT
ALGEBRAIC GEOMETRY SEMINAR

“On Lines on smooth quartics”

by

ALEXANDER DEGTYAREV

Abstract: (a never
ending project with I. Itenberg and S. Sertoz) It is a common understanding
that, thanks to the global Torelli theorem and the surjectivity of the period
map, any reasonable  question
concerning the topology and geometry of K3-surfaces can be reduced to a certain
arithmetical problem. We tried to apply this ideology to the study of the
possible configurations of straight lines on a nonsingular quartic surface in
P3. According to C. Segre, a nonsingular quartic in P3 may contain at most 64
lines, and one explicit example of a surface with exactly 64 lines is known.
The original proof, using classical algebraic geometry in the Italian school
style, is very complicated. We managed to reprove Segre's result using the  contemporary arithmetical approach. In
addition, we prove that, up to projective equivalence, a nonsingular quartic
with 64 lines is unique. Furthermore, we show that a real nonsingular quartic may
contain at most 56 real lines and, conjecturally, such a quartic is also unique
(although the latter statement is not quite definite yet). Alas, the proof is
transparent but heavily computer aided, the principal achievement being a stage
at which my laptop can handle it in finite time (although a human still
cannot).

Date:  21 September 2012,  Friday

Time: 15.40

Place: Bilkent,
Mathematics Seminar Room SA-141

Tea and cookies will be served before the
talk.

You are most cordially invited.