Counting Lines on Quartic Surfaces

TL;DR

This paper establishes the maximum number of lines on complex projective quartic surfaces as 64, explores line configurations on special non-K3 quartics, and provides examples of singular quartics with numerous lines.

Contribution

It proves the sharp bound of 64 lines on non-ruled complex quartic surfaces and analyzes line configurations on specific non-K3 quartics with examples.

Findings

Maximum of 64 lines on complex quartic surfaces proven

Line configurations studied on non-K3 quartics

Examples of singular quartics with many lines provided

Abstract

We prove the sharp bound of at most 64 lines on complex projective quartic surfaces (resp. affine quartics) that are not ruled by lines. We study configurations of lines on certain non-K3 surfaces of degree four and give various examples of singular quartics with many lines.