turkmath.org

In 1943, B. Segre proved that a smooth quartic surface in the complex projective space cannot contain more than 64 lines. (The champion, so-called Schur's quartic, has been known since 1882.) Even though a gap was discovered in Segre's proof (Rams, Schütt, 2015), the claim is still correct; moreover, it holds over any field of characteristic other than 2 or 3. (In characteristic 3, the right bound seems to be 112.) At the same time, it was conjectured that not any number between 0 and 64 can occur as the number of lines in a quartic. We tried to attack the problem using the theory of K3-surfaces and arithmetic of lattices. This relatively simple reduction has lead us to an extremely difficult arithmetical problem. Nevertheless, the approach turned out quite fruitful: for the moment, we have a complete classification of smooth quartics containing more than 52 lines. As an immediate consequence of this classification, we have the following: -- an alternative proof of Segre's bound 64; -- Shur's quartic is the only one with 64 lines; -- a real quartic may contain at most 56 real lines; -- a real quartic with 56 real lines is also unique; -- the number of lines takes values {0,...,52,54,56,60,64}. Conjecturally, we have a complete list of all quartics with more than 48 lines; there are about two dozens of species, most projectively rigid. I will discuss methods used in the proof and a few problems that are still open, e.g., the minimal fields of definition, triangle-free configurations, lines in singular quartics, etc. This subject is a joint work in progress with Ilia Itenberg and Sinan Sertöz.