JournalsjemsVol. 15, No. 2pp. 539–594

Welschinger invariants of small non-toric Del Pezzo surfaces

  • Ilia Itenberg

    Université de Strasbourg, France
  • Viatcheslav Kharlamov

    Université de Strasbourg, France
  • Eugenii Shustin

    Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv, Israel
Welschinger invariants of small non-toric Del Pezzo surfaces cover

Abstract

We give a recursive formula for purely real Welschinger invariants of the following real Del Pezzo surfaces: the projective plane blown up at~qq real and s1s \leq 1 pairs of conjugate imaginary points, where q+2s5q+2s\le 5, and the real quadric blown up at s1s \leq 1 pairs of conjugate imaginary points and having non-empty real part. The formula is similar to Vakil's recursive formula \cite{Va} for Gromov–Witten invariants of these surfaces and generalizes our recursive formula~\cite{IKS3} 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.

Cite this article

Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin, Welschinger invariants of small non-toric Del Pezzo surfaces. J. Eur. Math. Soc. 15 (2013), no. 2, pp. 539–594

DOI 10.4171/JEMS/367