Counting lines on surfaces, especially quintics

TL;DR

This paper develops a method using rational functions to count lines on surfaces, proving that smooth quintic surfaces in characteristic zero contain at most 127 lines and each line meets at most 28 others, with examples showing sharpness.

Contribution

Introduces rational functions on surfaces to count lines and establishes sharp bounds for lines on smooth quintic surfaces in characteristic zero.

Findings

Maximum of 127 lines on smooth quintic surfaces

Any line meets at most 28 others

Constructed examples demonstrating sharpness of bounds

Abstract

We introduce certain rational functions on a smooth projective surface X in IP^3 which facilitate counting the lines on X. We apply this to smooth quintics in characteristic zero to prove that they contain no more than 127 lines, and that any given line meets at most 28 others. We construct examples which demonstrate that the latter bound is sharp.