A nonsingular real plane curve of degree 9 with the maximal number of real components (an M-curve) has 29 components: one pseudo-line and 28 ovals, and the most intricate configurations are deep nests, where an oval contains a second oval which in turn contain a third. The source paper (Fiedler-Le Touzé, J. London Math. Soc. 2009) settled the sub-family of deep nests without outer ovals on the prohibition side; Orevkov (Progr. Math. 296, 2012) later completed that sub-family on the construction side and proved that every such curve must have an O1-jump. The problem posed here is the remainder: classify, for every arrangement ⟨J ⊔ α ⊔ 1⟨β ⊔ 1⟨γ⟩⟩⟩ with at least one outer oval (α ≥ 1, β ≥ 0, γ ≥ 1, α+β+γ = 26), whether it is realizable by a real algebraic M-curve of degree 9. The settled α = 0 sub-family is recorded here as background; the only refinement task remaining for it is the optional rigid-isotopy question described below. This is the currently most active open sector of the classification of real schemes of plane curves of degree 9 — the lowest degree for which Hilbert's 16th problem (part I) remains open.

Background

A real plane algebraic curve is the zero set of a real homogeneous polynomial in the real projective plane RP²; if the polynomial is nonsingular of degree m, Harnack's theorem bounds the number of connected components (all ovals except when m is odd, where exactly one component is a pseudo-line, i.e. a one-sided circle meeting every line) by g+1, where g=(m−1)(m−2)/2g=(m-1)(m-2)/2 is the genus. An M-curve attains the bound. The first half of Hilbert's 16th problem asks for the classification of the isotopy types of these curves in RP² of each degree, and of the possible mutual positions of their components. The classification is complete through degree 7, almost complete in degree 8, and wide open in degree 9 — the frontier. A real scheme records the mutual position of components in Viro's notation: ⟨J ⊔ α ⊔ 1⟨β⟩⟩ means the pseudo-line J, α empty ovals (ovals containing no other oval), and one oval containing β ovals; nested brackets iterate, so ⟨J ⊔ α ⊔ 1⟨β ⊔ 1⟨γ⟩⟩⟩ describes a depth-3 nest (for degree 9, called a deep nest): one oval — the outer oval O₁ of the nest — contains β empty ovals and one further oval, the median oval O₂, which in turn contains γ ovals; the α remaining ovals are exterior (empty) ovals lying outside O₁. For an M-curve of degree 9, g = 28, so there are 28 ovals plus J, forcing α+β+γ = 26 with γ ≥ 1 (the 28 ovals are: the α exterior ones, O₁, the β empty ones inside O₁, O₂, and the γ ones inside O₂); Bézout-type arguments (lines meet the curve in 9 points, conics in 18) imply all other ovals are empty and nest depth is at most 3. By 2009 the classification of deep-nest schemes with no outer ovals (α = 0) had reached the following state: Korchagin had prohibited β ∈ {1,2,3,25} (1986) and built new degree-9 M-curves by smoothing singular points (1992); Fiedler-Le Touzé's cited paper prohibited the ten remaining odd-(β,γ) schemes and constrained the complex schemes of the even ones. A complex scheme refines a real scheme for a dividing curve (Type I: the complexification separates the Riemann surface, inducing an orientation of the real locus preserved by complex conjugation): each oval carries a sign ± (orientation induced from one half), and pairs of injectively conjugate ovals carry a secondary sign Π ∈ {+,−}. The Rokhlin–Mishachev formula and Orevkov's formulas for curves with deep nests constrain the numbers of ovals of each sign; the complex scheme is a rigid isotopy invariant (invariant under isotopy through nonsingular real curves). A further rigid invariant for the deep-nest family is the O1-jump: a defect in the distribution of the tangency orders between a pencil of lines through a point on the outer oval O₁ and the curve, measured against the values forced by the complex orientation. The existence of an O1-jump (Fiedler-Le Touzé's Definition 1, developed from Fiedler's classical theorem on cubic pencils) is defined for every α, but the refined distribution tuple of her Definition 2 is defined only for curves with no outer ovals (α = 0), where the cyclic ordering of its underlying Lemma 1 essentially uses that hypothesis. The 2009 paper combined this cubic-pencil machinery with Orevkov's complex-orientation formulas (Topology 1999), the Orevkov–Viro congruence mod 8 for degree 9, and Korchagin's smoothing constructions. Later literature added link-theoretic obstructions: colored signatures and a Murasugi–Tristram-type inequality (Orevkov 2012, building on Gilmer–Orevkov's plumbing-diagram signatures), which turn out to be strictly stronger than the cubic-pencil toolbox on this family, and Orevkov's 2021 inequalities that first exposed a degree-9 phenomenon — a complex scheme realizable by symplectic (pseudoholomorphic) curves but not by algebraic ones — showing that degree 9 genuinely requires new restriction principles beyond those sufficient in degrees ≤ 8.

Problem Statement

Determine the complete classification of deep-nest real schemes of M-curves of degree 9 with at least one outer oval: for every triple of integers with α ≥ 1, β ≥ 0, γ ≥ 1, α+β+γ = 26, decide whether the real scheme ⟨J ⊔ α ⊔ 1⟨β ⊔ 1⟨γ⟩⟩⟩ is realizable by a nonsingular real algebraic curve of degree 9 in RP² with 29 real components (J is the pseudo-line; α counts the exterior empty ovals; β counts the empty ovals inside the outer oval O₁ of the nest but outside the median oval O₂; γ counts the ovals inside O₂). The sub-family α = 0 is settled and is recorded as background, not re-asked: realizable exactly for even β with 4 ≤ β ≤ 22 (Fiedler-Le Touzé 2009; Orevkov 2012), and every realization has an O1-jump (Orevkov 2012, Theorem 7.2). The open core is the set of 325 schemes with at least one outer oval, α ≥ 1 — for which only two prohibitions (⟨J ⊔ 2 ⊔ 1⟨1 ⊔ 1⟨23⟩⟩⟩, ⟨J ⊔ 3 ⊔ 1⟨1 ⊔ 1⟨22⟩⟩⟩), the β = 0 restriction to α odd and at least 7, and fifteen scattered constructions (seven algebraic; eight only pseudoholomorphic, whose algebraization is open) are known. A finer rigid-isotopy refinement is optional additional information: for the α = 0 family, where the distribution tuple of Definition 2 applies, the open question is whether the first (1₋) complex-scheme form of Theorem 2(1) can also occur for some β; for α ≥ 1 the cited definitions do not define an O1-jump distribution. For each such triple, realizability is meant in the algebraic sense: the scheme is realizable when some nonsingular real algebraic curve of degree 9 in RP² realizes it, and non-realizable when no such curve exists; the realizing curve must be a genuine real algebraic curve of degree 9, not a merely pseudoholomorphic one.

References

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