112 lines on smooth quartic surfaces (characteristic 3)

TL;DR

This paper proves that over fields of characteristic 3, smooth quartic surfaces in projective 3-space cannot have more than 112 lines, with the maximal case being the Fermat quartic, and establishes a characteristic-free upper bound for lines on quadrics on such surfaces.

Contribution

It establishes the maximum number of lines on smooth quartic surfaces in characteristic 3 and relates the extremal case to the Fermat quartic, providing a new upper bound.

Findings

No smooth quartic surface in characteristic 3 has more than 112 lines.

The Fermat quartic is projectively equivalent to the surface with 112 lines.

A characteristic-free upper bound for lines met by a quadric on a smooth quartic surface is derived.

Abstract

Over a field k of characteristic 3, we prove that there are no smooth quartic surfaces S in IP^3 with more than 112 lines. Moreover, the surface with 112 lines is projectively equivalent over k-bar to the Fermat quartic. As a key ingredient, we derive a characteristic free upper bound for the number of lines met by a quadric on a smooth quartic surface.