Relative enumerative invariants of real nodal del Pezzo surfaces

TL;DR

This paper introduces a signed count of real rational curves on specific real del Pezzo surfaces with a unique (-2)-curve, proving its invariance under deformations and point constraints.

Contribution

It defines a new enumerative invariant for real nodal del Pezzo surfaces and proves its invariance under certain deformations.

Findings

The signed count is independent of point constraints.

The count remains invariant under deformations preserving the real structure.

The invariant applies to surfaces with specific negativity conditions on the canonical class.

Abstract

The surfaces considered are real, rational and have a unique smooth real $(-2)$-curve. Their canonical class $K$ is strictly negative on any other irreducible curve in the surface and $K^2>0$. For surfaces satisfying these assumptions, we suggest a certain signed count of real rational curves that belong to a given divisor class and are simply tangent to the $(-2)$-curve at each intersection point. We prove that this count provides a number which depends neither on the point constraints nor on deformation of the surface preserving the real structure and the $(-2)$-curve.