Logarithmic equivalence of Welschinger and Gromov–Witten invariants

The Welschinger numbers, a kind of a real analogue of the Gromov–Witten numbers that count the complex rational curves through a given generic collection of points, bound from below the number of real rational curves for any generic collection of real points. Logarithmic equivalence of sequences is understood to mean the asymptotic equivalence of their logarithms. Such an equivalence is proved for the Welschinger and Gromov–Witten numbers of any toric Del Pezzo surface with its tautological real structure, in particular, of the projective plane, under the hypothesis that all, or almost all, the chosen points are real. A study is also made of the positivity of Welschinger numbers and their monotonicity with respect to the number of imaginary points.