Counting lines on quartic surfaces: new techniques and results

Abstract.

In the last years, two independent research teams (the one based in Ankara, Turkey, the other in Hannover, Germany) have tackled the problem of counting lines on smooth quartic surfaces; the former aimed at a complete classification using computer algebra system GAP, the latter strove for more geometrical insight.

The synergy between the two methods has fostered new ideas towards three goals: (1) finding a proof of the fact that the maximal number of lines is 64 which does not involve the flecnodal divisor; (2) proving the uniqueness of the surface with 64 lines with a geometrical approach; (3) adapting the methods to the K3 quartic case.

I will report about the state of the art.