Welschinger invariants of real del Pezzo surfaces of degree ≥ 2
Résumé
We compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree ≥ 2. We show that under some conditions, for such a surface X and a real nef and big divisor class D ∈ Pic(X), through any generic collection of −DK X − 1 real points lying on a connected component of the real part RX of X one can trace a real rational curve C ∈ |D|. This is derived from the positivity of appropriate Welschinger invariants. We furthermore show that these invariants are asymptotically equivalent, in the logarithmic scale, to the corresponding genus zero Gromov–Witten invariants. Our approach consists in a conversion of Shoval–Shustin recursive formulas counting complex curves on the plane blown up at seven points and of Vakil's extension of the Abramovich–Bertram formula for Gromov–Witten invariants into formulas computing real enumerative invariants.