Irreducible Rational Curve of Degree 10

• Irreducible rational curves of degree 10 are non-degenerate algebraic curves in projective spaces, birational to P¹ with defining irreducible equations.
• Explicit constructions, such as those exhibiting twelve ordinary triple points, demonstrate the use of degeneration and parametrization techniques in verifying classical enumerative predictions.
• The study links Severi varieties, moduli space dimensions, and toric structures to advance our understanding of deformation theory and algebraic geometry frameworks.

An irreducible rational curve of degree 10 is a non-degenerate algebraic curve of degree 10 in a projective space or surface, birational to the projective line $\PP^1$, and whose defining homogeneous equation is irreducible. In the complex projective plane $\PP^{2}$, the locus of such curves forms a fundamental component of the classical Severi variety, with deep connections to enumerative geometry, deformation theory, and the structure of Hilbert and Chow schemes. Recent advances provide explicit parametrizations, affirmative irreducibility results, and detailed descriptions in both toric and classical settings.

1. Foundational Properties and Irreducibility Criteria

In the classical setting, let $V_{d,0} \subset |\mathcal{O}_{\PP^2}(d)|$ denote the Severi variety of irreducible rational plane curves of degree $d$. For every $d \geq 1$, $V_{d,0}$ is nonempty, smooth, and irreducible of dimension $3d - 1$ (Zahariuc, 2017). For degree $10$, this yields a $29$-dimensional irreducible variety,

$\dim V_{10,0} = 3 \cdot 10 - 1 = 29.$

Irreducibility is established via induction and geometric degeneration arguments, specifically degenerating $\PP^{2}$0 to the union of a plane and a Hirzebruch surface intersecting along a line, then analyzing the central fiber via topological profiles and “doppelgänger” constructions. This structural theorem ensures that parametric families of irreducible rational degree-10 curves in $\PP^{2}$1 form a connected, smooth moduli space of expected dimension.

On rational surfaces $\PP^{2}$2, the locus $\PP^{2}$3 of morphisms $\PP^{2}$4 mapping with cycle class $\PP^{2}$5 (where $\PP^{2}$6 is the hyperplane class) is also irreducible and of expected dimension $\PP^{2}$7, computed as $\PP^{2}$8 (Dores, 2020).

2. Explicit Example: Rational Degree-10 Curve with 12 Triple Points

A concrete construction of an irreducible rational curve of degree 10 in $\PP^{2}$9 having twelve ordinary triple points was given by Orevkov (Orevkov, 12 Jan 2026), providing an explicit counterexample to a conjecture on the maximal number of triple points possible for rational curves of this degree. The curve is parametrized by

$V_{d,0} \subset |\mathcal{O}_{\PP^2}(d)|$0

The associated homogeneous equation $V_{d,0} \subset |\mathcal{O}_{\PP^2}(d)|$1 has total degree 10 (the polynomial is given explicitly in (Orevkov, 12 Jan 2026)). The genus calculation confirms rationality: $V_{d,0} \subset |\mathcal{O}_{\PP^2}(d)|$2 Irreducibility follows from the existence of the rational parametrization: the coordinate ring admits no decomposition, so the curve is geometrically irreducible.

Each of the twelve singularities is an ordinary triple point (analytically isomorphic to $V_{d,0} \subset |\mathcal{O}_{\PP^2}(d)|$3). Three lie on the line $V_{d,0} \subset |\mathcal{O}_{\PP^2}(d)|$4 and nine arise from a 3-fold covering construction. The ordinary nature of each triple point is verified analytically via local normal forms.

3. Families and Degenerations: Analytic and Geometric Aspects

The aforementioned example extends to a one-parameter analytic family $V_{d,0} \subset |\mathcal{O}_{\PP^2}(d)|$5 of degree-10 curves, such that for $V_{d,0} \subset |\mathcal{O}_{\PP^2}(d)|$6, $V_{d,0} \subset |\mathcal{O}_{\PP^2}(d)|$7 is irreducible and has twelve ordinary triple points, and as $V_{d,0} \subset |\mathcal{O}_{\PP^2}(d)|$8 the curve degenerates to the union of the nine lines of the dual Hesse arrangement (which already contains twelve triple points) and an additional line $V_{d,0} \subset |\mathcal{O}_{\PP^2}(d)|$9: $d$0 This degeneration is constructed by the "bubbling" lemma (Mikhalkin–Orevkov): explicit perturbations of intersection points allow for the persistence and recombination of singularities into a single irreducible component for $d$1. The technique leverages sequences of blow-up/blow-down and explicit control of parameter shifts, guaranteed by analytic implicit function arguments (Orevkov, 12 Jan 2026).

4. Moduli, Severi Varieties, and Enumerative Geometry

Enumeratively, the Severi variety $d$2 parametrizes irreducible, degree-10 rational curves in $d$3, corresponding to configurations with the maximal number $d$4 nodes. However, explicit counts of such curves through $d$5 general points are governed by the recursive Caporaso–Harris formula; for $d$6, this yields a finite number $d$7, computable algorithmically (Zahariuc, 2017).

The moduli space setting is further illuminated by the construction of the space of morphisms $d$8, which is irreducible of dimension $d$9, related to $d \geq 1$0 via the cycle-class map and birational geometry (Dores, 2020). Geometric properties of the image (such as the types and quantities of singularities) distinguish strata within this broader space.

5. Rational Degree-10 Curves in Toric and Higher-Dimensional Settings

In the context of projective toric varieties $d \geq 1$1, every degree-10 rational curve can be constructed via a degree-10 Cayley structure, i.e., a suitable affine-linear map $d \geq 1$2 defined combinatorially on a face $d \geq 1$3 of $d \geq 1$4 (Ilten et al., 2023). Smoothness, irreducibility, and degree criteria for the resulting curve $d \geq 1$5 are completely combinatorial: $d \geq 1$6 must be a "primitive" and "smooth" Cayley structure.

For $d \geq 1$7, irreducible components of the Hilbert scheme $d \geq 1$8 whose general member is a smooth rational degree-10 curve are in bijection with coordinate $d \geq 1$9-planes (for $V_{d,0}$0). The dimension of such a component is $V_{d,0}$1, and each corresponds to a maximal Cayley structure. The normalization of the closure of the torus orbit of such a curve in the Chow variety is a toric variety determined by explicit hyperplane arrangements in the moment polytope.

The table below summarizes key data for loci of rational degree-10 curves in $V_{d,0}$2 (Ilten et al., 2023):

Parameter	Value (for $V_{d,0}$3-plane in $V_{d,0}$4)	Description
Number of families	$V_{d,0}$5	Distinct irreducible components
Dimension	$V_{d,0}$6	Dimension of the component for $V_{d,0}$7-plane
Chow normalization	Toric variety for Cayley fan	Normalization of corresponding torus orbit

6. Irreducibility in Families and Expected Dimension

The irreducibility results generalize to the setting of morphisms from $V_{d,0}$8 to a rational surface $V_{d,0}$9, with the locus $3d - 1$0 irreducible of dimension $3d - 1$1 when the divisor class $3d - 1$2 contains at least one smooth rational curve (Dores, 2020). The same deformation-theoretic and cycle-class arguments as in the projective plane establish this, and the expected dimension formula

$3d - 1$3

specializes to $3d - 1$4 in the case $3d - 1$5. Each irreducible component is dominated by an open set consisting of parametrizations birational onto their image, giving a unique geometric structure to the moduli space.

7. Singularities, Pathologies, and Counterexamples

The existence of an irreducible rational curve of degree 10 in $3d - 1$6 with exactly twelve ordinary triple points (and no higher-order singularities) disproves a prior conjecture bounding the number of such singularities for rational curves of given degree (Orevkov, 12 Jan 2026). The elementary calculation

$3d - 1$7

validates rationality. The explicit analytic and geometric constructions of such curves, as well as the control over their singularities under deformation, further demonstrate the rich structure within the irreducible components of the Severi variety for degree 10. All singularities are shown to be analytically standard, locally reducible to $3d - 1$8.

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References:

(Dores, 2020, Zahariuc, 2017, Orevkov, 12 Jan 2026, Ilten et al., 2023)