A decade of line counting: an overview

Abstract: I will give a brief overview of a long project that started a decade ago (in collaboration with Ilia Itenberg and Sinan Sert\"oz and in parallel with S{\l}awomir Rams and Matthias Sch\"utt) and originally intended to bridge a minor gap in the proof of Segre's celebrated theorem on 64 lines on a smooth quartic surface. Confining ourselves to polarized K3-surfaces, now we manage to answer questions that no one even dared to ask, mostly because of lack of tools. For example, we

$\bullet$ obtained sharp upper bounds on the possible number of lines on a smooth polarized K3-surface of any degree,

$\bullet$ obtained similar bounds for quartics, sextics, and octics with singularities,

$\bullet$ advanced in the understanding of conics on K3-surfaces,

$\bullet$ started the study of twisted cubics.

I will try to discuss both classical (more than 5 years old) results and recent advances; if time permits, I will also try to outline the techniques used.

algebraic geometry

Audience: researchers in the discipline

ODTU-Bilkent Algebraic Geometry Seminars