Exponential rarefaction of real curves with many components
Publications Mathématiques de l'IHÉS, Tome 113 (2011), pp. 69-96.

Given a positive real Hermitian holomorphic line bundle L over a smooth real projective manifold X, the space of real holomorphic sections of the bundle L d inherits for every d∈ℕ a L 2-scalar product which induces a Gaussian measure. When X is a curve or a surface, we estimate the volume of the cone of real sections whose vanishing locus contains many real components. In particular, the volume of the cone of maximal real sections decreases exponentially as d grows to infinity.

@article{PMIHES_2011__113__69_0,
     author = {Gayet, Damien and Welschinger, Jean-Yves},
     title = {Exponential rarefaction of real curves with many components},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {69--96},
     publisher = {Springer-Verlag},
     volume = {113},
     year = {2011},
     doi = {10.1007/s10240-011-0033-3},
     zbl = {1227.32028},
     mrnumber = {2805598},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-011-0033-3/}
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Gayet, Damien; Welschinger, Jean-Yves. Exponential rarefaction of real curves with many components. Publications Mathématiques de l'IHÉS, Tome 113 (2011), pp. 69-96. doi : 10.1007/s10240-011-0033-3. http://archive.numdam.org/articles/10.1007/s10240-011-0033-3/

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