# 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

## 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 $s≤1$ pairs of conjugate imaginary points, where $q+2s≤5$, and the real quadric blown up at $s≤1$ pairs of conjugate imaginary points and having non-empty real part. The formula is similar to Vakil's recursive formula [22] for Gromov–Witten invariants of these surfaces and generalizes our recursive formula [12] 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