Genus-one rational first integrals and intrinsic dicritical bounds in the effective Poincaré problem for foliations of the projective plane

A singular holomorphic foliation of the complex projective plane may be algebraically integrable, given by the level curves of a rational function called a rational first integral. The Poincaré problem asks for an effective upper bound on the degree of such an integral, and its genus-refined form bounds the degree in terms of the genus of the integral and intrinsic data of the foliation. Galindo and Monserrat (2014) settled the case with one dicritical exceptional divisor, and — excluding genus one — gave degree bounds and decision algorithms for integrals of any fixed genus when the number of dicritical exceptional divisors is at most two, extending to at least three dicritical divisors only when an auxiliary set of independent invariant curves is supplied in advance. The follow-up literature confirms that both boundaries persist: the genus-one (elliptic pencil) case with at least two dicritical divisors still has neither an intrinsic degree bound nor a decision algorithm in this dicritical framework — only its non-isotrivial subcase is bounded, by results of Pereira–Svaldi and Cousin–Lins Neto–Pereira — and no general construction of the required auxiliary invariant-curve sets is known. The surviving open problem is the genus-one completion together with an intrinsic treatment of foliations with at least three dicritical divisors.

Background

Let $\mathbb{P}^2$ be the complex projective plane. A singular holomorphic foliation $\mathcal{F}$ of degree $r$ on $\mathbb{P}^2$ is a codimension-one foliation with finitely many singular points, defined by a homogeneous polynomial 1-form of degree $r+1$; the degree $r$ counts the tangencies of $\mathcal{F}$ with a general projective line. The foliation is algebraically integrable when it admits a rational first integral, i.e. a nonconstant rational function $f=P/Q$ whose level curves are leaves of $\mathcal{F}$; the invariant algebraic curves of $\mathcal{F}$ are then the members of the pencil of plane curves $P-\lambda Q$, and the genus of the first integral is the geometric genus (genus of the normalization) of a general member of this pencil. A pencil is called isotrivial when all its smooth members are isomorphic curves; for genus-one pencils this is equivalent to a constant $j$-invariant.

The classical Poincaré problem asks for an effective bound on the degree $d$ of the first integral (or of the general invariant curve) in terms of $r$ and other intrinsic data of $\mathcal{F}$; Painlevé's companion problem asks to recognize or bound the genus $g$ of the general invariant curve. For foliations with only nondegenerate singularities Poincaré proved the inequality $d(r-4)\le 4(g-1)$. Cerveau and Lins Neto (1991) and Carnicer (1994) settled the generic and the nondicritical cases, the latter giving $d\le r+2$ when no singularity on the invariant curve is dicritical in the following sense. After resolving the singularities of $\mathcal{F}$ by point blow-ups, each exceptional divisor is either invariant under the lifted foliation or transverse to it; a transverse exceptional divisor is called a dicritical divisor, and $\mathrm{dic}(\mathcal{F})$ denotes the number of dicritical exceptional divisors in the minimal resolution. Dicritical divisors are exactly the places where a rational first integral can acquire arbitrarily large degree, so they are the obstruction to Carnicer-type bounds and the central object of the dicritical approach.

Lins Neto (2002) proved that the unrestricted problem is impossible: he constructed families of algebraically integrable foliations with fixed degree, fixed local analytic type and fixed number of singularities whose first integrals have arbitrarily large degree and arbitrarily large genus. Any positive result must therefore refine the data, and two refinements are known. First, Pereira (2002) bounded $d$ for foliations of general type in terms of $g$, $r$ and the plurigenera $h^0(\mathbb{P}^2,K_{\mathcal{F}}^{\otimes m})$, where $K_{\mathcal{F}}$ is the canonical sheaf of the foliation; Pereira and Svaldi (2019) made this effective for birationally fibred foliations of bounded genus, and, combining their Proposition 8.2 with Cousin, Lins Neto and Pereira (2022, Proposition 4.2 — the fibers of a non-isotrivial elliptic fibration lie in the zero set of a section of $K_{\mathcal{F}}^{\otimes 12}$, giving degree at most $12(r-1)$ on $\mathbb{P}^2$), also bounded the non-isotrivial genus-one case. The isotrivial genus-one case is not covered by any of these results: there, the same authors prove only that some invariant curve of degree at most $6(r-1)$ exists, which does not bound the degree of the general member of the pencil. Second, Galindo and Monserrat (2014) proved, working on the surface obtained by blowing up the dicritical configuration and using orthogonality relations in its Néron–Severi lattice (the group of divisor classes modulo numerical equivalence): for $\mathrm{dic}(\mathcal{F})=1$ the bound $d\le(r+2)^2/4$ together with an algorithm deciding algebraic integrability, with no restriction on the genus (Theorem 1); for $\mathrm{dic}(\mathcal{F})\le 2$ and $g\ne 1$ a bound on $d$ depending only on $g$, $r$ and the local analytic type of the dicritical singularities, together with an algorithm deciding existence of a first integral of any prescribed genus $g\ne 1$ and computing it (Theorem 2); and the analogous statements for $\mathrm{dic}(\mathcal{F})\ge 3$ under the additional hypothesis that a set of $\mathrm{dic}(\mathcal{F})-2$ independent algebraic solutions is known, where independent means that the classes of these invariant curves, together with one canonical-class element and the classes of the non-dicritical exceptional divisors, are linearly independent over the rationals in the Néron–Severi group of the resolution surface, and where the bound then also depends on the degrees and blow-up multiplicities of those auxiliary curves (Theorem 3). The same authors had earlier settled the case in which the cone of curves of the dicritical-resolution surface — the convex cone in the real Néron–Severi space spanned by the classes of effective curves — is polyhedral, a finiteness condition satisfied for instance when the blow-up involves at most eight points: there, algebraic integrability can be decided and the first integral computed from intrinsic data alone (Galindo and Monserrat 2006). The auxiliary-set hypothesis of Theorem 3 is the device that replaces this finiteness assumption when the cone need not be polyhedral. The genus enters this method exclusively through the adjunction formula, which relates the genus of a curve to the intersection of its divisor class with the canonical class; at $g=1$ the genus contribution $2(g-1)$ vanishes, so the formula carries no information and every genus-refined statement in this framework is restricted to $g\ne 1$. This restriction survives verbatim in the authors' 2025 follow-up, which still takes a restricted set of independent algebraic solutions as input.

Problem Statement

Let $\mathcal{F}$ be a singular holomorphic foliation of degree $r$ on the complex projective plane $\mathbb{P}^2$, and let $\mathrm{dic}(\mathcal{F})$ denote the number of dicritical exceptional divisors in the minimal resolution of its singularities.

(A) Genus-one completion. For foliations with $\mathrm{dic}(\mathcal{F})\ge 2$, solve the following two sub-questions, each of which is a well-posed problem in its own right: (A1) determine whether the degree of a rational first integral of genus one (a pencil of curves of geometric genus one, isotrivial or not) admits an upper bound depending only on intrinsic data of $\mathcal{F}$ — its degree $r$ and the local analytic type of its dicritical singularities; and (A2) construct an algorithm which, given a defining homogeneous 1-form of $\mathcal{F}$ together with the resolution of its dicritical singularities, decides whether $\mathcal{F}$ has a rational first integral of genus one and computes it in the affirmative case. The degree-bound sub-question (A1) is open precisely for the isotrivial genus-one pencils, since the non-isotrivial case is bounded by elliptic-fibration methods; the decision algorithm (A2) is open for all genus-one first integrals with $\mathrm{dic}(\mathcal{F})\ge 2$, because the adjunction-based method of Galindo and Monserrat degenerates at $g=1$.

(B) Intrinsic treatment for $\mathrm{dic}(\mathcal{F})\ge 3$. For foliations with $\mathrm{dic}(\mathcal{F})\ge 3$, determine whether the decision procedure and the degree bound for first integrals of prescribed genus $g\ne 1$ can be made to depend only on the foliation's degree, the genus and its singularity data, with no externally supplied invariant curves, beyond the polyhedral cone-of-curves case where intrinsic data already suffice.

The common scientific objective linking (A) and (B) is a complete and intrinsic effective solution of the genus-refined, dicritical-divisor-based Poincaré problem: from the foliation alone (defining equation and singularity data), decide algebraic integrability in every fixed genus — including the missing genus one — and produce degree bounds depending only on the stated intrinsic data. Both components are the two remaining obstructions to that single objective in the source framework, and a complete solution requires both: settling either component alone leaves the other open. The unrestricted problem — a bound depending only on $r$ — is excluded from the question, since Lins Neto proved by explicit families that no such bound exists.

References

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