Modular inequalities and Alexander polynomials of pencil type conic-line arrangements

Technical Summary: Modular Inequalities and Alexander Polynomials of Pencil Type Conic-Line Arrangements

Problem Statement
The paper addresses the computation of Alexander polynomials, $\Delta_C(t)$, for specific classes of complex projective plane curves known as pencil-type conic-line arrangements. These are curves whose irreducible components are lines and smooth conics, realized as the union of fibers from a pencil of degree $d \ge 3$ curves. A central open problem in the field is whether the Alexander polynomial is determined solely by the combinatorics of the arrangement (specifically, the weak combinatorial type). While this "combinatoriality" conjecture fails for general plane curves (evidenced by Zariski pairs), it holds for hyperplane arrangements. The author investigates whether a similar combinatorial determination holds for a specific subclass of conic-line arrangements, particularly those arising from pencils of degree 3 curves.

Methodology
The author employs a combination of topological and algebraic techniques to bound and compute the multiplicities of roots of the Alexander polynomial:

1. Modular Inequalities: The paper utilizes recent results (specifically from [11]) regarding modular inequalities for curve complements. These inequalities provide upper bounds for the multiplicities ($b_k$) of roots of the Alexander polynomial that are powers of primes. The bound is given by the dimension of the first cohomology of a generalized Aomoto complex, denoted $\beta_p$, which is a combinatorial invariant derived from the weak combinatorial type of the curve.
   $$b_k \le \beta_p$$
   where $k = p^s$.

2. Lower Bounds via Fiber-Type Structures: For curves admitting quasi-fiber-type structures, lower bounds for these multiplicities are established using results from [11], specifically Theorem 8, which states that for a union of $s$ members of a pencil, the multiplicity of a primitive root of order $k$ (where $k|s$) is at least $s-2$.

3. Linear System Analysis: To compute the combinatorial invariant $\beta_p$, the author constructs a linear system $S$ over the field $\mathbb{Z}_p$. The variables correspond to the irreducible components of the curve. The equations in $S$ are derived from the resonance condition $\omega \wedge \omega_1 = 0$ in the generalized Aomoto complex, utilizing the intersection data of the curve's singularities (ordinary and $A_3$ type).

4. Local Alexander Polynomials: The author uses the divisibility property of the Alexander polynomial (Theorem 9) and explicit formulas for local Alexander polynomials of ordinary singularities and $A_3$ singularities (Lemma 10) to filter possible roots.

Key Contributions and Results

• Combinatorial Determination for Degree 3 Pencils:
  The primary result (Theorem 1) establishes that for pencil-type conic-line arrangements arising from a pencil of degree 3 curves with transversal base points, the multiplicity of primitive roots of unity of odd order in $\Delta_C(t)$ is combinatorial. Furthermore, if the arrangement has at most two non-ordinary singularities, the multiplicity of roots of order $2^k$ is also combinatorial.

• Explicit Formulas for Alexander Polynomials:
  The paper provides explicit formulas for the Alexander polynomial $\Delta_C(t)$ for arrangements formed by the union of $m$ fibers ($3 \le m \le 6$) of a degree 3 pencil, assuming ordinary singularities at base points (Theorem 2).

  • $m=3$: $\Delta_C(t) = (t-1)^{s-1}(t^2+t+1)$, except for the Ceva arrangement where the factor is squared.
  • $m=4$: $\Delta_C(t) = (t-1)^{s-1}(t+1)^{b_2}(t^2+1)^{b_4}$ with $b_2, b_4 \ge 2$. Under specific conditions (at most two non-ordinary singularities), $b_2=b_4=2$.
  • $m=5$: $\Delta_C(t) = (t-1)^{s-1}(t^4+t^3+t^2+t+1)^3$.
  • $m=6$: $\Delta_C(t) = (t-1)^{s-1}(t+1)^{b_2}(t^2+t+1)^4(t^2-t+1)^{b_6}$ with $b_2, b_6 \ge 4$. Under specific conditions, $b_2=4$.

• Refined Analysis for $m=4$:
  The paper extends the results for $m=4$ to cases with more non-ordinary singularities. Theorem 3 and Theorem 20 show that if either there are at most three non-ordinary singularities, or there exists a base point through which an odd number of lines pass, then the exponents are exactly $b_2=b_4=2$, yielding $\Delta_C(t) = (t-1)^{s-1}(t+1)^2(t^2+1)^2$.

• Abstract Combinatorial Types and Realizability:
  The author identifies a specific "abstract" weak combinatorial type $W$ (Theorem 22) where every base point singularity involves an even number of lines. For this type, the combinatorial invariant is $\beta_2(W) = 3$, suggesting a potential multiplicity $b_2=3$ (contrasting with the $b_2=2$ found in realizable cases satisfying the conditions of Theorem 3). The paper poses the open question (Question 24) whether this abstract type $W$ is realizable by an actual complex plane curve and, if so, whether distinct Alexander polynomials can arise from the same combinatorial type.

Significance and Claims
The paper claims to demonstrate a "distinctive phenomenon" where, for a specific subclass of pencil-type conic-line arrangements, the Alexander polynomial is at least partially combinatorial. This contrasts with the general failure of combinatoriality for plane curves (due to Zariski pairs).

The author emphasizes that the techniques employed—specifically the use of modular inequalities to bound root multiplicities and the analysis of the generalized Aomoto complex—are suitable for broader use and may be applied to compute Alexander polynomials for more general classes of curves. The work bridges the gap between the combinatorial theory of hyperplane arrangements and the more complex topology of conic-line arrangements, providing concrete formulas where previously only bounds or open questions existed. The paper modestly notes that while the modular inequality becomes an equality in the studied cases, this is not true in general, and the identified abstract type $W$ serves as a boundary case for understanding the limits of combinatorial determination.