Lines on K3 quartic surfaces in characteristic 3

TL;DR

This paper studies the maximum number of lines on K3 quartic surfaces in characteristic 3, establishing bounds and explicit examples, and characterizing the Fermat quartic surface as extremal.

Contribution

It proves bounds on the number of lines on K3 quartic surfaces in characteristic 3 and provides explicit examples and classifications of extremal cases.

Findings

Surfaces with 112 lines are projectively equivalent to the Fermat quartic.

Maximum lines on such surfaces are at most 67, reduced to 58 if a star configuration exists.

Explicit equations for families with 58 lines and a surface with 48 lines are given.

Abstract

We investigate the number of straight lines contained in a K3 quartic surface \(X\) defined over an algebraically closed field of characteristic 3. We prove that if \(X\) contains 112 lines, then \(X\) is projectively equivalent to the Fermat quartic surface; otherwise, \(X\) contains at most 67 lines. We improve this bound to 58 if \(X\) contains a star (ie four distinct lines intersecting at a smooth point of \(X\)). Explicit equations of three 1-dimensional families of smooth quartic surfaces with 58 lines, and of a quartic surface with 8 singular points and 48 lines are provided.