Two kinds of real lines on real del Pezzo surfaces of degree 1

TL;DR

This paper classifies real lines on real del Pezzo surfaces of degree 1 into elliptic and hyperbolic types using an intrinsic Pin-structure, showing invariance under automorphisms and deformations, with a fixed difference of 16 between their counts.

Contribution

It introduces a new intrinsic Pin-structure to distinguish line types and proves the invariance of the splitting and the fixed difference in their counts.

Findings

Lines split into elliptic and hyperbolic types based on the Pin-structure

The splitting remains invariant under automorphisms and deformations

The difference between the number of hyperbolic and elliptic lines is always 16

Abstract

We show how the real lines on a real del Pezzo surface of degree 1 can be split into two species, elliptic and hyperbolic, via a certain distinguished, intrinsically defined, Pin-structure on the real locus of the surface. We prove that this splitting is invariant under real automorphisms and real deformations of the surface, and that the difference between the total numbers of hyperbolic and elliptic lines is always equal to 16.