Welschinger invariant and enumeration of real plane rational curves

Itenberg, I.; Kharlamov, V.; Shustin, E.

Welschinger invariant and enumeration of real plane rational curves

Authors

Itenberg, I.

Kharlamov, V.

Shustin, E.

Description

Welschinger's invariant bounds from below the number of real rational curves through a given generic collection of real points in the real projective plane. We estimate this invariant using Mikhalkin's approach which deals with a corresponding count of tropical curves. In particular, our estimate implies that, for any positive integer $d$, there exists a real rational curve of degree $d$ through any collection of $3d-1$ real points in the projective plane, and, moreover, asymptotically in the logarithmic scale at least one third of the complex plane rational curves through a generic point collection are real. We also obtain similar results for curves on other toric Del Pezzo surfaces.
17 pages, LATEX2e; revised version: some statements modified (the case of rational geometrically ruled surfaces is replaced by that of the toric Del Pezzo surfaces), proofs extended, 2 figures added