Combined count of real rational curves of canonical degree 2 on real del Pezzo surfaces with $K^2=1$

TL;DR

The paper introduces two invariant counting methods for real rational curves of canonical degree 2 on real del Pezzo surfaces with K^2=1, resulting in counts of 30 and 96 that are independent of surface choice.

Contribution

It develops two new intrinsic sign systems for counting such curves, ensuring invariance and providing explicit counts for different divisor classes.

Findings

Total count of 30 for all canonical degree 2 divisor classes.

Count of 96 when excluding class -2K and summing over real structures.

Counts are invariant under surface deformation.

Abstract

We propose two systems of "intrinsic" signs for counting such curves. In both cases the result acquires an exceptionally strong invariance property: it does not depend on the choice of a surface. One of our counts includes all divisor classes of canonical degree 2 and gives in total 30. The other one excludes the class $-2K$, but adds up the results of counting for a pair of real structures that differ by Bertini involution. This count gives 96.