The maximum number of lines lying on a K3 quartic surface

TL;DR

This paper proves an upper bound of 64 lines on certain K3 quartic surfaces with isolated rational double points, extending previous theorems and providing new insights into line configurations without using flecnodal divisors.

Contribution

It establishes a new maximum number of lines on K3 quartic surfaces with specific singularities, extending classical results and introducing a novel proof approach.

Findings

Maximum of 64 lines on the specified quartic surfaces

Extended Segre--Rams--Schütt theorem to new cases

Constructed examples of non-smooth K3 quartic surfaces with many lines

Abstract

We show that there cannot be more than 64 lines on a quartic surface admitting isolated rational double points over an algebraically closed field of characteristic $p \neq 2,\,3$, thus extending Segre--Rams--Sch\"utt theorem. Our proof offers a deeper insight into the triangle-free case and takes advantage of a special configuration of lines, thereby avoiding the technique of the flecnodal divisor. We provide several examples of non-smooth K3 quartic surfaces with many lines.