Lines on quartic surfaces

TL;DR

This paper determines the maximum number of real lines on nonsingular spatial quartic surfaces and classifies those with more than 52 lines, showing all such quartics are projectively rigid.

Contribution

It establishes the maximum number of lines on real nonsingular quartic surfaces and provides a complete classification of quartics with over 52 lines.

Findings

Maximum of 64 real lines on nonsingular spatial quartic surfaces

Complete classification of quartics with more than 52 lines

Any number up to 52 lines can be realized by some quartic

Abstract

We show that the maximal number of (real) lines in a (real) nonsingular spatial quartic surface is 64 (respectively, 56). We also give a complete projective classification of all quartics containing more than 52 lines: all such quartics are projectively rigid. Any value not exceeding 52 can appear as the number of lines of an appropriate quartic.