Welschinger invariant and enumeration of real plane rational curves

Details Welschinger invariant and enumeration of real plane rational curves Welschinger invariant and enumeration of real plane rational curves

2003-03-30

arxiv.org/abs/math/0303378v2

Mathematics

Algebraic Geometry

17 pages, LATEX2e; revised version: some statements modified (the case of rational geometrically ruled surfaces is replaced by

Scientific paper

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.

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