Welschinger invariant and enumeration of real plane rational curves
arXiv:math/0303378
Abstract
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 , there exists a real rational curve of degree through any collection of 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