[Paper Review] On the topology of arrangements of a cubic and its inflectional tangents

Shinzo Bannai, Benoît Guerville-Ballé+2 more|arXiv (Cornell University)|Jul 26, 2016

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper constructs Zariski pairs among $k$-Artal arrangements—comprised of a smooth cubic and $k$ inflectional tangents—for $k = 3,4,5,6$ by analyzing subarrangements and topological invariants. It proves that such pairs exist and can be distinguished geometrically by the number of collinear triples among singular points on the cubic, with Type I ($k-3$ collinear triples) and Type II ($k-4$ collinear triples) arrangements forming non-homeomorphic pairs despite identical combinatorics.

ABSTRACT

A $k$-Artal arrangement is a reducible algebraic curve composed of a smooth cubic and $k$ inflectional tangents. By studying the topological properties of their subarrangements, we prove that for $k=3,4,5,6$, there exist Zariski pairs of $k$-Artal arrangements. These Zariki pairs can be distinguished in a geometric way by the number of collinear triples in the set of singular points contained in the cubic.

Motivation & Objective

• To construct Zariski pairs among reducible plane curves formed by a smooth cubic and its inflectional tangents for $k=3,4,5,6$.
• To establish a geometric invariant capable of distinguishing the topology of such arrangements when combinatorics are identical.
• To demonstrate that the number of collinear triples among singular points on the cubic serves as a topological invariant for $k$-Artal arrangements.
• To extend the method of subarrangement invariants—such as $D_6$-covers, Alexander polynomials, splitting numbers, and linking sets—to detect non-homeomorphic configurations.

Proposed method

• Define $k$-Artal arrangements as the union of a smooth cubic $E$ and $k$ inflectional tangents $L_{P_i}$ at flex points $P_i$.
• Introduce subarrangement invariants $\tilde{\Phi}_{E,\mathcal{L}_J}^{D_6}$, $\tilde{\Phi}_{E,\mathcal{L}_J}^{\mathrm{Alex}}$, $\tilde{\Phi}_{E,\mathcal{L}_J}^{\mathrm{split}}$, and $\tilde{\Phi}_{E,\mathcal{L}_J}^{\mathrm{lks}}$ restricted to 3-element subcollections of tangents.
• Use known results: $D_6$-cover existence and Alexander polynomial depend on collinearity of three flex points; splitting number and linking set vary accordingly.
• Define Type I ($k-3$ collinear triples) and Type II ($k-4$ collinear triples) arrangements to classify configurations.
• Prove that the number of such collinear triples is preserved under homeomorphisms preserving the cubic, hence distinguishing topological types.
• Apply the subarrangement method: if invariants differ on subarrangements, the full arrangements cannot be homeomorphic, yielding Zariski pairs.

Experimental results

Research questions

• RQ1 For $k=3,4,5,6$, do there exist $k$-Artal arrangements with identical combinatorics but non-homeomorphic topological types in $\mathbb{P}^2$?
• RQ2 Can the number of collinear triples among the singular points of a $k$-Artal arrangement serve as a topological invariant?
• RQ3 Are there geometric invariants—beyond fundamental groups or Alexander polynomials—that can distinguish Zariski pairs in reducible plane curves?
• RQ4 Can the subarrangement method, using restricted invariants on 3-tangent subsets, detect non-homeomorphism in $k$-Artal arrangements?
• RQ5 Why do no Zariski pairs exist for $k=1,2,7,8,9$ among $k$-Artal arrangements?

Key findings

• For $k=3$, a Zariski pair exists and is distinguished by the collinearity of the three flex points: if collinear, certain invariants (e.g., $D_6$-cover existence) take value 1, otherwise 0.
• For $k=4,5,6$, $k$-Artal arrangements of Type I (with $k-3$ collinear triples) and Type II (with $k-4$ collinear triples) yield Zariski pairs when combinatorially equivalent.
• The number of 3-element subarrangements with collinear flex points is exactly $k-3$ for Type I and $k-4$ for Type II, and this count is preserved under homeomorphisms fixing the cubic.
• The invariants $\tilde{\Phi}_{E,\mathcal{L}_J}^{D_6}$, $\tilde{\Phi}_{E,\mathcal{L}_J}^{\mathrm{Alex}}$, $\tilde{\Phi}_{E,\mathcal{L}_J}^{\mathrm{split}}$, and $\tilde{\Phi}_{E,\mathcal{L}_J}^{\mathrm{lks}}$ restricted to 3-tangent subarrangements take distinct values depending on collinearity, enabling topological distinction.
• The proof relies on the fact that a homeomorphism preserving the cubic must induce a bijection on subarrangements, but the differing counts of collinear triples break this, proving non-homeomorphism.
• For $k=1,2,7,8,9$, no such Zariski pairs exist among $k$-Artal arrangements, as shown by combinatorial and topological constraints.