64 lines on smooth quartic surfaces

TL;DR

This paper proves that smooth quartic surfaces in projective three-space over fields not of characteristic 2 or 3 can have at most 64 lines, establishing a sharp bound on line intersections.

Contribution

It establishes the maximum number of lines on smooth quartic surfaces and derives the precise bound that each line meets at most 20 others.

Findings

No smooth quartic surface has more than 64 lines.

Any line on such a surface meets at most 20 other lines.

The bound is sharp and optimal.

Abstract

Let k be a field of characteristic other than 2,3. We prove that there are no geometrically smooth quartic surfaces in IP^3 with more than 64 lines. As a key step, we derive the sharp bound that any line meets at most 20 other lines on a smooth quartic.