Smooth Surfaces with Maximal Lines

TL;DR

This paper establishes an upper bound on the number of lines on smooth degree d surfaces in projective 3-space and characterizes those surfaces that attain this maximum, linking them to specific algebraic equations.

Contribution

It provides a sharp upper bound for the number of lines on smooth surfaces and characterizes the extremal cases explicitly.

Findings

Maximum of d^2(d^2-3d+3) lines on such surfaces

Surfaces with maximum lines are defined by specific equations involving p^e+1 powers

Characterization of extremal surfaces in terms of algebraic equations

Abstract

We prove that a smooth projective surface of degree $d$ in $\mathbb P^3$ contains at most $d^2(d^2-3d+3)$ lines. We characterize the surfaces containing exactly $d^2(d^2-3d+3)$ lines: these occur only in prime characterize $p$ and, up to choice of projective coordinates, are cut out by equations of the form $x^{p^{e}+1}+y^{p^{e}+1}+z^{p^{e}+1}+ w^{p^{e}+1} = 0.$