On large configurations of lines on quartic surfaces

TL;DR

This paper investigates the maximum number of lines on non-K3 quartic surfaces, establishing bounds based on singularities and describing unique configurations that attain these bounds.

Contribution

It provides new upper bounds on the number of lines on non-K3 quartic surfaces and characterizes the configurations that achieve these bounds.

Findings

Non-K3 quartic surfaces with isolated double points have at most 20 lines.

Surfaces with certain singularities can have up to 31 lines.

Unique configurations attain these maximum bounds.

Abstract

We estimate the number of lines on a non-K3 quartic surface. Such a surface with only isolated double point(s) contains at most twenty lines; this bound is attained by a unique configuration of lines and by a surface with a certain limited set of singularities. We have similar itemized bounds for other types of non-simple singularities, which culminate in at most 31 lines on a non-K3 quartic not ruled by lines; this bound is only attained on the quartic monoids described by K.~Rohn.