Welschinger invariants of small non-toric Del Pezzo surfaces

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 $s\le 1$ pairs of conjugate imaginary points, where $q+2s\le 5$, and the real quadric blown up at $s\le 1$ 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.

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References in corpus (2)

• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• Optimalité, congruences et calculs d'invariants des variétés symplectiques réelles de dimension quatre

Cited by in corpus (1)

• The open Gromov-Witten-Welschinger theory of blowups of the projective plane