Welschinger invariants of real Del Pezzo surfaces of degree $\ge 3$

TL;DR

This paper introduces a recursive formula for real Welschinger invariants of certain Del Pezzo surfaces, including modifications for specific cases, and establishes their positivity and asymptotic behavior compared to Gromov-Witten invariants.

Contribution

It provides the first recursive computation method for real Welschinger invariants of degree ≥ 3 Del Pezzo surfaces, including new modifications for degree 3 cases.

Findings

Proves positivity of the invariants.

Establishes asymptotic equivalence to genus zero Gromov-Witten invariants.

Shows invariants are congruent modulo 4.

Abstract

We give a recursive formula for purely real Welschinger invariants of real Del Pezzo surfaces of degree $K^2\ge 3$, where in the case of surfaces of degree $3$ with two real components we introduce a certain modification of Welschinger invariants and enumerate exclusively the curves traced on the non-orientable component. As an application, we prove the positivity of the invariants under consideration and their logarithmic asymptotic equivalence, as well as congruence modulo $4$, to genus zero Gromov-Witten invariants.