A bound for the number of lines lying on a nonsingular surface in $3$-space over a finite field

TL;DR

This paper establishes an upper bound on the number of lines lying on a nonsingular surface of degree d in three-dimensional projective space over a finite field, with the bound being optimal for specific degrees.

Contribution

It provides a new bound on the maximum number of lines on nonsingular surfaces in finite fields, extending understanding of geometric configurations in algebraic geometry.

Findings

Bound of ((d-1)q+1)d lines for nonsingular degree d surfaces

Bound is optimal for d=2, d=√q+1, d=q+1

Advances knowledge of line configurations on algebraic surfaces

Abstract

A nonsingular surface of degree $d \geq 2$ in $\mathbb{P}^3$ over $\mathbb{F}_q$ has at most $((d-1)q+1)d$ $\mathbb{F}_q$-lines, and this bound is optimal for $d = 2, \sqrt{q}+1, q+1$.