Symmetries and equations of smooth quartic surfaces with many lines

TL;DR

This paper explicitly constructs smooth complex quartic surfaces with numerous lines, analyzes their symmetries and configurations, and addresses a question about the Fermat quartic's determinantal presentation.

Contribution

It provides explicit equations for quartic surfaces with many lines and explores their automorphisms and line configurations, including a solution to Oguiso's question.

Findings

Explicit equations for quartic surfaces with over 52 lines

Analysis of automorphisms and line configurations

Determinantal presentation of the Fermat quartic confirmed

Abstract

We provide explicit equations of some smooth complex quartic surfaces with many lines, including all 10 quartics with more than 52 lines. We study the relation between linear automorphisms and some configurations of lines such as twin lines and special lines. We answer a question by Oguiso on a determinantal presentation of the Fermat quartic surface.