Welschinger invariants of small non-toric Del Pezzo surfaces

TL;DR

This paper introduces a recursive formula for real Welschinger invariants of certain small non-toric Del Pezzo surfaces, extending previous formulas and proving positivity and asymptotic behavior.

Contribution

It provides a new recursive formula for real Welschinger invariants of specific non-toric Del Pezzo surfaces, generalizing prior work on toric cases.

Findings

Recursive formula for real Welschinger invariants

Proof of positivity of these invariants

Asymptotic equivalence to Gromov-Witten invariants

Abstract

We give a recursive formula for purely real Welschinger invariants of the following real Del Pezzo surfaces: the projective plane blown up at $q$ real and $s \leq 1$ pairs of conjugate imaginary points, where $q+2s\le 5$, and the real quadric blown up at $s \leq 1$ pairs of conjugate imaginary points and having non-empty real part. The formula is similar to Vakil's recursive formula for Gromov-Witten invariants of these surfaces and generalizes our recursive formula for purely real Welschinger invariants of real toric Del Pezzo surfaces. As a consequence, we prove the positivity of the Welschinger invariants under consideration and their logarithmic asymptotic equivalence to genus zero Gromov-Witten invariants.