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RANDOM REAL BRANCHED COVERINGS OF THE PROJECTIVE LINE

Part of: Holomorphic functions of several complex variables Curves Limit theorems

Published online by Cambridge University Press:  09 February 2021

Michele Ancona Open the ORCID record for Michele Ancona [Opens in a new window]
Michele Ancona*
Affiliation:
Tel Aviv University, School of Mathematical Sciences (michi.ancona@gmail.com)
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Abstract

In this paper, we construct a natural probability measure on the space of real branched coverings from a real projective algebraic curve $(X,c_X)$ to the projective line $(\mathbb{C} \mathbb {P}^1,\textit{conj} )$ . We prove that the space of degree d real branched coverings having “many” real branched points (for example, more than $\sqrt {d}^{1+\alpha }$ , for any $\alpha>0$ ) has exponentially small measure. In particular, maximal real branched coverings – that is, real branched coverings such that all the branched points are real – are exponentially rare.

Keywords

real algebraic curvesbranched coveringsrandom mapsBergman kernel

MSC classification

Primary: 14P99: None of the above, but in this section
Secondary: 14H30: Coverings, fundamental group 60F10: Large deviations 32A25: Integral representations; canonical kernels (Szegő, Bergman, etc.)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 5 , September 2022 , pp. 1783 - 1799
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

The author is supported by the Israeli Science Foundation through ISF Grants 382/15 and 501/18

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