Lines on smooth K3-surfaces

Abstract:

I will settle a conjecture (originally based on the classification of smooth models of singular K3-surfaces) on the maximal number of lines in a sextic surface in $\mathbb P^4$ or octic surface in $\mathbb P^3$ (36 lines; the same bound is sharp for triquadrics). As in the case of spatial quartics, a classification of sextics and octics with many lines is also obtained. For example, there are several families of triquadrics with 32 lines other than the classical Kummer family (constructed via quadratic line complexes).

I will also discuss other polarizations of K3-surfaces. The asymptotic maximum is 24 lines, all lines lying in the fibers of an elliptic pencil. This bound is sharp for infinitely many polarizations. In the original three cases of quartics in $\mathbb P^3$, sextics in $\mathbb P^4$, and octics in $\mathbb P^5$, the line maximizing polarized K3-surfaces are also discriminant minimizing singular K3-surfaces admitting a smooth model. (It was this observation, valid in several known cases, together with the classification of small smooth models that gave rise to the conjecture in the first place.) However, this tendency does not persist: it fails already for surfaces of degree ten in $\mathbb P^6$.

The principal tools are the theory of periods of K3-surfaces and Nikulin’s theory of discriminant forms.

This work is partially supported by TUBITAK project 116F211.