Random real branched coverings of the projective line

Revêtements ramifiés réels aléatoires de la droite projective

Michele Ancona (1)

1 AGL - Algèbre, géométrie, logique

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 $(\mathbb{C}\mathbb{P}^1,\textrm{conj})$. 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.

Domaines: Géométrie algébrique [math.AG]; Probabilités [math.PR]

Michele Ancona. Random real branched coverings of the projective line. 2020. ⟨hal-02434851⟩