Divergent separatrices of holomorphic foliations on the complex projective plane with a unique singular point: an explicit saddle-node example and rigorous divergence for the Alcántara–Mozo-Fernández nilpotent family

A holomorphic codimension-one foliation of degree $d$ on the complex projective plane $\mathbb{CP}^2$ has at most $d^2+d+1$ singularities counted with multiplicity (Milnor number). The 2023–2024 work of Alcántara and Mozo-Fernández studied the extremal strata in which this bound is attained at a single point, and posed three problems. This record concerns the two that remain open. First, find an explicit projective foliation of some degree whose only singularity is a saddle-node and whose weak separatrix is a genuinely divergent (non-convergent) formal power series. Second, prove rigorously that the formal separatrix of the explicit two-parameter nilpotent family constructed in that work (Theorem 4.2, odd degree $d>1$) diverges — numerical evidence from the first 50 series coefficients suggests divergence for all parameter values $\alpha,\beta\neq 0$, but no proof exists; even the degree-3 instance — the case explicitly requested in the source — is open, and would give the first example of a second-type foliation with a unique singular point and a divergent separatrix. The third problem posed in the source (existence of logarithmic, non-dicritical, quasi-homogeneous foliations with a unique singular point) was resolved negatively in 2025 by Fernández-Pérez, García Barroso, and Saravia-Molina under invariant algebraic curve hypotheses (reference 2, Section 7.4, Theorems 7.6 and 7.7), narrowing this record to the surviving divergent-separatrix core.

Background

A holomorphic codimension-one foliation $\mathcal{F}$ of degree $d$ on $\mathbb{CP}^2$ is given in homogeneous coordinates by a homogeneous polynomial 1-form $\omega$ of degree $d+1$ satisfying the integrability (Frobenius) condition $\omega\wedge d\omega=0$, up to multiplication by a homogeneous polynomial of degree $d-1$. Its singular points are the common zeros of the coefficients, and the Milnor number $\mu$ of an isolated singularity measures its complexity; for a foliation of degree $d$ the total Milnor number is at most $d^2+d+1$. This record concerns foliations where the maximum is attained at a single point, i.e. $\mu_p= d^2+d+1$ at the unique singular point $p$. A saddle-node is a singularity whose linear part has one zero eigenvalue and one non-zero eigenvalue; in suitable local coordinates $(y,z)$ it has the Dulac normal form $\omega = z^{k+1}\,dy - y(1+\lambda z^k)\,dz$; it has a strong separatrix $C_1=\{z=0\}$, always analytic, and a weak separatrix $C_2=\{y=0\}$, which in general exists only as a formal invariant curve (a formal power series solving the invariance equation) and may fail to converge. A nilpotent singularity is one whose linear part is nilpotent; the source paper constructs, for every odd $d>1$ and parameters $\alpha,\beta\in\mathbb{C}^*$, an explicit degree-$d$ foliation (Theorem 4.2) with a unique nilpotent singularity at $(1\!:\!0\!:\!0)$ of Milnor number $d^2+d+1$. A formal power series is Gevrey of order $s\ge 0$ if its $n$-th coefficient is $O(A^n(n!)^s)$; convergence is the case $s=0$, and Gevrey series of positive order are typically divergent but Borel summable along directions (Martinet–Ramis theory). A foliation is a generalized curve if, after any reduction of singularities, no saddle-node appears; it is of second type if, for every saddle-node appearing in the reduction process, the weak separatrix is not contained in any component of the exceptional divisor. Every generalized curve is of second type, but not conversely; second-type foliations form a family invariant under Cremona transformations, and their local analytic structure is governed by their invariant curves, so the convergence behavior of formal separatrices is the key obstruction to understanding them. A singularity is dicritical if the blow-up produces an exceptional component covered by leaves; a foliation is logarithmic if locally $\omega=f_1\cdots f_m\,\sum_i \lambda_i\, df_i/f_i$ with $\sum_i\lambda_i d_i=0$, where $d_i=\deg f_i$. Relevant classical tools include the Camacho–Sad index (whose sum along a separatrix is constrained by the exceptional divisor's self-intersection), the GSV and Baum–Bott indices, the Płoski inequality for the Milnor number of a curve singularity, and Cerveau's theorem that a closed logarithmic form with holomorphic coefficients must be exact.

Problem Statement

Two mutually reinforcing open problems from Alcántara–Mozo-Fernández (arXiv:2306.05719, Remark 4.3 and the discussion following Theorem 4.2). (1) Construct an explicit example of a projective foliation of some degree on $\mathbb{CP}^2$ whose only singular point is a saddle-node and whose weak separatrix is genuinely divergent (a formal, non-convergent power series). (2) For the explicit family of Theorem 4.2 — for odd $d>1$ and $\alpha,\beta\in\mathbb{C}^*$, the degree-$d$ foliation with unique nilpotent singularity at $(1\!:\!0\!:\!0)$ of Milnor number $d^2+d+1$ — prove that the formal separatrix $y=h(z)$, where $h(z)=z^{(d+1)/2}\cdot\tilde h(z)$ with $\tilde h(0)=\beta$ and $\tilde h(z)=W(t)$ under the substitution $t=z^{(d^2+1)/2}$, and $W(t)$ is the formal solution of the ordinary differential equation $(W(t)-\beta-\alpha t W(t)^d)\cdot\bigl[\tfrac{d^2+1}{2}\,t\,W'(t)+\tfrac{d-1}{2}\,W(t)\bigr]+(W(t)-\beta)^2=0$, is divergent. The source explicitly poses the case $d=3$ ('It would be good, for $d=3$, to find an explicit example of a foliation of this type with a non-convergent separatrix, but up to now, we don't know it'), while its numerical experiments with the first 50 coefficients of $W$ support divergence for every $\alpha,\beta\neq 0$ throughout the family. The outcome sought is a second-type foliation with a unique singular point and a genuinely divergent separatrix (a foliation whose unique singular point has non-zero linear part is of second type for $d>2$); problems (1) and (2) are posed independently, neither implying the other. Within (2), establishing divergence in the case $d=3$ — the instance the source explicitly requests — already supplies the first such example, while the general family, for every odd $d>1$ and all $\alpha,\beta\in\mathbb{C}^*$, is the stronger full target.

References

1. Alcántara, Claudia R.; Mozo-Fernández, Jorge. 'Remarks on Foliations on CP² with a Unique Singular Point.' Bulletin of the Brazilian Mathematical Society, New Series 55(4), Paper No. 50 (2024). DOI: 10.1007/s00574-024-00426-3, https://doi.org/10.1007/s00574-024-00426-3; arXiv:2306.05719, https://arxiv.org/abs/2306.05719.
2. Fernández-Pérez, Arturo; García Barroso, Evelia R.; Saravia-Molina, Nancy. 'On some indices of foliations and applications.' Research in the Mathematical Sciences 13(1), Article 3 (online 12 December 2025, print March 2026). DOI: 10.1007/s40687-025-00587-7, https://doi.org/10.1007/s40687-025-00587-7; arXiv:2506.01090, https://arxiv.org/abs/2506.01090. (Section 7.4, Theorems 7.6 and 7.7, resolves negatively the source paper's conjecture on logarithmic non-dicritical quasi-homogeneous foliations with a unique singular point.)
3. Fernández, Percy; Puchuri, Liliana; Rosas, Rudy. 'Foliations on P² with only one singular point.' Geometriae Dedicata 216(5), Article 59 (online August 2022, print October 2022). DOI: 10.1007/s10711-022-00724-4, https://doi.org/10.1007/s10711-022-00724-4; arXiv:2103.00537, https://arxiv.org/abs/2103.00537. (Proves that foliations on $\mathbb{P}^2$ with only one singular point of algebraic multiplicity one have no invariant algebraic curve, and presents degree-3 examples.)
4. Mozo-Fernández, Jorge; Pantaleón-Mondragón, P. Rubí. 'Projective Foliations with a Unique Singular Point and Maximum Algebraic Multiplicity.' arXiv preprint (August 2026). arXiv:2608.06588, https://arxiv.org/abs/2608.06588. (Follow-up by the same research group; studies the opposite stratum $A_{d,d}$ where the algebraic multiplicity is maximal.)
5. Alcántara, Claudia R.; Pantaleón-Mondragón, Rubí. 'Foliations on CP² with a unique singular point without invariant algebraic curves.' Geometriae Dedicata 207(1), 193–200 (online 2019, print 2020). DOI: 10.1007/s10711-019-00492-8, https://doi.org/10.1007/s10711-019-00492-8. (The authors' 2019 companion paper that constructed the nilpotent examples.)
6. Fernández-Pérez, Arturo; Oliveira Sagnori Bernardes, Vângellis. 'Holomorphic Foliations of Degree Four on the Complex Projective Space.' Bulletin of the Brazilian Mathematical Society, New Series 56(2), Article 22 (online April 2025, print June 2025). DOI: 10.1007/s00574-025-00444-9, https://doi.org/10.1007/s00574-025-00444-9. (Concerns foliations on $\mathbb{P}^n$ for $n\ge 3$.)