Welschinger invariants of small non-toric Del Pezzo surfaces

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.