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Article Dans Une Revue International Mathematics Research Notices Année : 2003

Welschinger invariant and enumeration of real plane rational curves

Résumé

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.

Dates et versions

hal-00082526 , version 1 (28-06-2006)

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Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin. Welschinger invariant and enumeration of real plane rational curves. International Mathematics Research Notices, 2003, num. 49, pp.2639-2653. ⟨10.1155/S1073792803131352⟩. ⟨hal-00082526⟩
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