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Oberseminar WiSe 12/13 Zusammenfassung des Vortrags von Alexander Degtyarev (Bilkent)

On lines on smooth quartics

(a never ending joint 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).