On curves of degree 10 with 12 triple points

This paper, titled "On Curves of Degree 10 with 12 Triple Points" by S. Yu. Orevkov, provides a constructive counterexample to a conjecture in algebraic geometry and explores the existence of specific families of singular plane curves.

1. The Problem and Context

The paper addresses a conjecture by Dimca, Harbourne, and Sticlaru, which suggested that there is no reduced plane curve of degree $d > 9$ whose components are all rational and whose singularities (including infinitely near points) all have multiplicity 3.

Additionally, the author responds to a question by Joaquim Roé regarding the existence of an irreducible rational curve of degree 10 in $\mathbb{CP}^2$ possessing 12 ordinary triple points. Such a curve is significant because its existence challenges established bounds on the relationship between a curve's degree and its singularity types.

2. Key Contributions and Results

The author provides three primary mathematical constructions:

• Proposition 1(a): An Irreducible Rational Curve. The author constructs an irreducible rational curve of degree 10 with 12 ordinary triple points. This is provided via a specific polynomial parametrization and a Cartesian equation. This result serves as the direct counterexample to the aforementioned conjecture.
• Proposition 1(b): A Union of Three Quartics. The author constructs a reducible curve consisting of the union of three rational quartics that together possess 19 ordinary triple points. This demonstrates that the bound of 9 in the original conjecture should be raised to at least 12.
• Proposition 2: An Analytic Family of Curves. The author proves the existence of an analytic family of curves $C_u$ of degree 10. For $u \neq 0$, the curves are irreducible with 12 triple points, and as $u \to 0$, the family degenerates into the union of the dual Hesse arrangement (9 lines with 12 triple points) and an additional line.

3. Methodology

A. Construction of the Degree 10 Curve (Symmetry-Based)

To find the curve in Proposition 1(a), Orevkov uses a reduction method via birational transformations:

1. Symmetry Reduction: He searches for a polynomial of the form $f(x, y, z) = f_1(x, y^3, z)$, effectively working on a weighted projective plane $\mathbb{P}^2_{1,3,1}$ or a Hirzebruch surface $\Sigma_3$.
2. Geometric Transformation: By blowing up the triple points and blowing down specific fibers, the problem is transformed into finding smooth curves on $\Sigma_0 = \mathbb{P}^1 \times \mathbb{P}^1$ belonging to specific linear systems ($|A+3B|$ and $|3A+B|$) that satisfy specific tangency conditions.
3. Parametrization: He uses a symmetric approach where the two curves are related by a diagonal swap, allowing for a direct calculation of the parametrization.

B. Construction of the Three Quartics (Cremona Transformation)

For Proposition 1(b), the author employs a Cremona quadratic transformation. By starting with a specific quartic and applying the transformation relative to chosen base points (including points of cubic tangency), he reduces the complex problem to a configuration of a cuspidal cubic, a conic, and four lines, which is uniquely determined up to automorphism.

C. Perturbation of the Dual Hesse Arrangement

To prove Proposition 2, the author uses a "bubbling" technique:

1. Initial Perturbation: He starts with the dual Hesse arrangement and a generic line $L_0$. He constructs a parametrization that "smooths" the intersections of $L_0$ with the other lines while attempting to preserve the triple points.
2. Preserving Triple Points: He defines a system of linear equations to ensure that the triple points do not split into nodes during the perturbation.
3. Implicit Function Theorem: After showing that a solution exists up to $O(u^2)$ using a specific parameter vector $b_a$, he applies the Implicit Function Theorem to prove the existence of a full analytic family where all 12 triple points are strictly preserved.

4. Significance

The paper is significant for several reasons:

• Refutation of Conjectures: It provides concrete counterexamples to existing conjectures regarding the maximum degree of curves with specific singularity types.
• Methodological Innovation: The approach of constructing perturbations in parametric form rather than Cartesian form is noted as a potentially useful technique for more general settings in algebraic geometry.
• Computational Rigor: The author provides a recursive method for computing power series expansions of these curves, bridging theoretical geometry with computational algebraic methods (using tools like Maple).