Lines in supersingular quartics

TL;DR

This paper determines the maximum number of lines on supersingular quartic surfaces over fields of characteristics 2 and 3, providing bounds and classifications, and compares these to non-supersingular cases.

Contribution

It establishes precise bounds on the number of lines in supersingular quartic surfaces and classifies large line configurations, advancing understanding of their geometric structure.

Findings

Supersingular quartics in characteristic 2 have at most 32 lines, or exactly 40.

Supersingular quartics in characteristic 3 have at most 52 lines, or exactly 112.

Non-supersingular quartics have at most 60 lines.

Abstract

We show that the number of lines contained in a supersingular quartic surface is 40 or at most 32, if the characteristic of the field equals 2, and it is 112, 58, or at most 52, if the characteristic equals 3. If the quartic is not supersingular, the number of lines is at most 60 in both cases. We also give a complete classification of large configurations of lines.