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Welschinger invariant and enumeration of real plane rational curves

I. Itenberg·V. Kharlamov·E. Shustin
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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 $d$, 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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I. Itenberg, V. Kharlamov, E. Shustin. 2003-06-16. Welschinger invariant and enumeration of real plane rational curves. https://arxiv.org/abs/math/0303378

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