turkmath.org

It is classically known that, although each plane or quadric contains infinitely many straight lines and each smooth cubic surface contains exactly 27 lines, a generic surface of degree 4 or higher has no lines at all. Nevertheless, some surfaces do have lines, and one can ask such natural questions as the maximal number of lines, the possible configurations, etc. We will consider the first nontrivial case of smooth quartic surfaces. Starting with the classical results and examples (Schur’s quartic, Segre’s bound, etc.), we will discuss the recent developments and limitless prospective generalizations, viz. • very few configurations/quartics with many lines (more than 52), • the dependence on the field (C vs. R vs. Q; the latter is still open), • the case of positive characteristic and supersingular quartics. If time permits, we will also consider a few related topics. Thus, one of the latest developments is the fact that quartics with many lines turn out to be smooth spatial models of singular K3-surfaces of small discriminant, and the champion —Schur’s quartic carrying 64 lines— can be characterized as the “smallest” one. Along these lines, we can also construct new examples and suggest conjectures to the wide-open problem of counting lines in some other classical projective models of K3-surfaces; for example, we conjecture that a smooth plane sextic curve may have at most 72 tritangents. (For comparison, recall that a smooth quartic curve always has exactly 28 bitangents—a fact closely related to the 27 lines in a cubic surface.)