Birational Geometry of Special Quotient Foliations and Chazy's Equations

TL;DR

The paper identifies three special quotient foliations $\mathcal{F}_3$, $\mathcal{F}_4$, and $\mathcal{F}_6$ on rational surfaces, each arising as a quotient of a linear foliation by a cyclic group of orders $3$, $4$, and $6$, tied to invariant nodal curves with self-intersection $C^2=3,2,1$. It then relates these foliations to the reduced Chazy IV, V, VI equations by constructing explicit degree-two plane models $\mathcal{H}_3$, $\mathcal{H}_4$, $\mathcal{H}_6$ and establishing birational equivalences to foliations on $\mathbb{P}(1,2,3)$ induced by the Chazy equations, with explicit symmetry actions generating $S_3$, $\mathbb{Z}_2$, and trivial groups, respectively. The work also provides a non-degenerate plane model $\mathcal{J}$ for Brunella’s foliation, explains its quartic de Jonquières symmetry, and presents a detailed factorization of the quartic into standard Cremona maps, illustrating how the foliated flop is realized. Additionally, the authors analyze quotients of linear foliations by the standard Cremona involution, giving explicit plane models (notably Cayley’s nodal cubic) and the quotients of degree-one foliations, along with a thorough study of the birational automorphism groups of these special quotient foliations. The results yield new plane models, explicit symmetry descriptions for the Chazy equations, and a cohesive birational framework connecting special foliations, Cremona transformations, and Chazy dynamics with potential applications to the broader study of foliations on rational surfaces and their moduli.

Abstract

The works of Brunella and Santos have singled out three special singular holomorphic foliations on projective surfaces having invariant rational nodal curves of positive self-intersection. These foliations can be described as quotients of foliations on some rational surfaces under cyclic groups of transformations of orders three, four, and six, respectively. Through an unexpected connection with the reduced Chazy IV, V and VI equations, we give explicit models for these foliations as degree-two foliations on the projective plane (in particular, we recover Pereira's model of Brunella's foliation). We describe the full groups of birational automorphisms of these quotient foliations, and, through this, produce symmetries for the reduced Chazy IV and V equations. We give another model for Brunella's very special foliation, one with only non-degenerate singularities, for which its characterizing involution is a quartic de Jonquières one, and for which its order-three symmetries are linear. Lastly, our analysis of the action of monomial transformations on linear foliations poses naturally the question of determining planar models for their quotients under the action of the standard quadratic Cremona involution; we give explicit formulas for these as well.

Figures (18)

• Figure 1: Configuration of the invariant curves for the foliations in Theorems \ref{['thm:f3']}, \ref{['thm:f4']} and \ref{['thm:f6']}. In blue, the nodal cubics that produce the links; in orange, an inflectional line; in green, a line, a conic and a nodal cubic, respectively. Numbers denote local intersection multiplicities with the nodal cubic.
• Figure 2: Birational equivalence between $\mathbb{P}(1,2,3)$ and $\mathbb{P}^2$. Numbers in square brackets denote the self-intersection of the corresponding curve.
• Figure 3: The cycles of rational curves and the effect on them of the automorphisms of order three, four and six, respectively.
• Figure 4: Brunella's foliated flop and its factorization.
• Figure 5: To the left, schematic minimal reduction of singularities of $\mathcal{G}_{\mathrm{IV}}$; to the right, a morphism to $\mathbb{P}^2$. Dotted lines represent curves that are not invariant by the foliation.

Theorems & Definitions (34)

• Theorem 1: Pereira
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10