On wall-crossing invariance of certain sums of Welschinger numbers

TL;DR

This paper extends the construction of real enumerative invariants, originally for low-degree curves on certain del Pezzo surfaces, to curves of any degree on all del Pezzo surfaces with degree up to 3, aiming for invariance under wall-crossing.

Contribution

It generalizes previous invariants to higher degrees and broader classes of del Pezzo surfaces, enhancing the understanding of real enumerative geometry.

Findings

Constructed invariants for all anti-canonical degrees on del Pezzo surfaces with K^2 ≤ 3.

Demonstrated invariance of these sums under wall-crossing.

Extended previous results from degree ≤ 2 to higher degrees and surfaces.

Abstract

We continue our quest for real enumerative invariants not sensitive to changing the real structure and extend the construction we uncovered previously for counting curves of anti-canonical degree $\leqslant 2$ on del Pezzo surfaces with $K^2=1$ to curves of any anti-canonical degree and on any del Pezzo surfaces of degree $K^2\leqslant 3$.