# Counting lines on projective surfaces

## TL;DR

This paper establishes a new upper bound on the number of lines on smooth degree-d surfaces in projective space, improving previous bounds and making some classical arguments rigorous, especially for degrees six and higher.

## Contribution

It provides the best known bound for the number of lines on such surfaces for degrees six and above, refining and rigorously justifying Segre's classical results.

## Key findings

- New upper bound on lines for degree-d surfaces with d ≥ 3
- Improves and rigorizes Segre's classical bounds
- Achieves the best known bounds for d ≥ 6

## Abstract

We prove a bound on the number of lines on a smooth degree-d surface in three-dimensional projective space for $d \geq 3$. This bound improves a bound due to Segre and renders some of his arguments rigorous. It is the best known bound for $d \geq 6$.