Lines on smooth K3surfaces
Alex Degtyarev (Bilkent University, Ankara)
What 


When 
Jun 28, 2018 from 03:30 pm to 04:30 pm 
Where  Mainz, 05432 (Hilbertaum) 
Add event to calendar 
vCal iCal 
Abstract:
I will settle a conjecture (originally based on the classification of smooth models of singular K3surfaces) 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 K3surfaces. 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 K3surfaces are also discriminant minimizing singular K3surfaces 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 K3surfaces and Nikulinâ€™s theory of discriminant forms.
This work is partially supported by TUBITAK project 116F211.