Hilbert modular foliations on the projective plane

Abstract

Abstract. We describe explicitly holomorphic singular foliations on the projective plane corresponding to natural foliations of Hilbert modular surfaces associated to the field Q( √ 5). These are concrete models for a very special class of foliations in the recent birational classification of foliations on projective surfaces.

Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI

“…Dans [26], Mendes et Pereira donnent les premiers exemples de modèles birationnels explicites pour des feuilletages modulaires. La découverte de ces feuilletages est fondée sur une bonne connaissance de la surface sousjacente.…”

“…La découverte de ces feuilletages est fondée sur une bonne connaissance de la surface sousjacente. Il apparaît que les structures transversalement projectives de deux de ces exemples (les feuilletages H 2 et H 3 associés à √ 5 dans [26]) correspondent à des déformations isomonodromiques de feuilletages de Riccati à quatre pôles sur P 1 ×P 1 → P 1 , c'est-à-dire à deux solutions de l'équation de Painlevé VI (PVI). Ces solutions sont, par construction, algébriques ; elles sont des transformées d'Okamoto de solutions icosahédrales de DubrovinMazzocco [15], les solutions n°31 et 32 de la liste de Boalch [5].…”

Transversely Projective Foliations on Surfaces: Existence of Minimal Form and Prescription of Monodromy

“…Both foliations admit transversely projective structures with reduced polar divisor whose support consists of a rational quintic and a line, cf. [6,11]. For H 2 the eccentricity is equal to 2 = 6 − (2 + 2) while for H 3 it is equal to 1 = 6 − (3 + 2).…”

The Noether–Fano inequalities for codimension one singular holomorphic foliations

“…Also in [13] there is a pair of modular foliations H 2 and H 3 , of degrees two and three resp obtained from H 5 and H 9 by taking quotient with their symmetry group. H 2 is not birationally equivalent to a linear foliation or a degree zero foliation (this can be proved directly by considering the leaves of such foliations or as consequence of the birational classification of [1]).…”

“…After the quotient, the involution which sends the horizontal discs to the vertical ones becomes a birational involution transforming one modular foliation into the other. In [13] there is an explicit description of a pair of modular foliations in the projective plane with degrees five and nine, denoted H 5 and H 9 . There is a degree five birational involution χ with H 9 = χ * (H 5 ) and χ is a composition χ = Q 3 • Q 2 • Q 1 of three standard Cremona transformations, with Ind( ; that H 8 has a point with l(p,…”