On the addition technique for Betti and Poincaré polynomials of plane curves

Technical Summary: On the Addition Technique for Betti and Poincaré Polynomials of Plane Curves

Problem Statement
The paper investigates the topological and algebraic properties of reduced plane curves in the complex projective plane $\mathbb{P}^2_{\mathbb{C}}$, specifically focusing on how the Poincaré and Betti polynomials behave under the "addition technique." This technique involves analyzing the union of two reduced curves, $C_1$ and $C_2$, where their intersection is 0-dimensional. The study is motivated by the classical theory of central hyperplane arrangements, where a well-known addition-deletion theorem (Theorem 1.1) relates the Poincaré polynomial of an arrangement to that of a sub-arrangement and a restriction. The author seeks to establish analogous polynomial identities for plane curves, particularly those possessing quasi-homogeneous singularities.

Methodology
The paper operates exclusively over the field of complex numbers ($\mathbb{C}$). The methodology relies on the following theoretical framework:

1. Singularities and Invariants: The study focuses on curves with quasi-homogeneous singularities. For such curves, the total Tjurina number $\tau(C)$ coincides with the total Milnor number $\mu(C)$. The paper utilizes the Jacobian ideal $J_f$ and the module of Jacobian syzygies to define the freeness of a curve and its exponents.
2. Polynomial Definitions:
   • The Poincaré polynomial of a reduced curve $C$ of degree $d$ is defined as $P(C; t) = 1 + (d-1)t + ((d-1)^2 - \tau(C))t^2$.
   • The Betti polynomial of the complement $M(C) = \mathbb{P}^2_{\mathbb{C}} \setminus C$ is defined as $B_{M(C)}(t) = 1 + (e-1)t + ((d-1)^2 - \mu(C) - d + e)t^2$, where $e$ is the number of irreducible components.
3. Addition Technique: The core analytical tool is the derivation of a relationship between the invariants of the union $C_1 \cup C_2$ and the individual curves $C_1$ and $C_2$. A crucial step involves Lemma 3.1, which establishes the behavior of the total Milnor number under addition: $\mu(C_1 \cup C_2) = \mu(C_1) + \mu(C_2) + 2c_1c_2 - r$, where $c_i$ are the degrees and $r$ is the number of intersection points.

Key Contributions and Results
The paper presents two main theorems providing polynomial identities for the addition of plane curves:

• Theorem 1.2 (Poincaré Polynomials): For two reduced curves $C_1, C_2$ such that their union admits only quasi-homogeneous singularities and their intersection consists of $r$ points, the Poincaré polynomial of the union satisfies:
  $$P(C_1 \cup C_2; t) = P(C_1; t) + P(C_2; t) + t - 1 + (r - 1)t^2$$
  This result holds under the specific assumption that $C_1 \cup C_2$ has only quasi-homogeneous singularities.

• Theorem 1.3 (Betti Polynomials): Under the same intersection conditions (0-dimensional intersection of $r$ points), the Betti polynomial of the complement of the union satisfies:
  $$B_{M(C_1 \cup C_2)}(t) = B_{M(C_1)}(t) + B_{M(C_2)}(t) + t - 1 + (r - 1)t^2$$
  Notably, this result for Betti polynomials does not explicitly require the quasi-homogeneous singularity assumption in its statement, relying instead on the general behavior of Milnor numbers and component counts.

• Applications to Free Curves: The paper demonstrates the utility of these identities through several examples (Examples 3.3–3.5) involving line arrangements and conics. By applying the addition formula, the author computes the Poincaré polynomial of the resulting union and observes that it splits over the rationals (e.g., into $(1+d_1t)(1+d_2t)$). This splitting serves as a heuristic indicator that the resulting curve is "free," a property that can then be verified using existing criteria (such as the du Plessis–Wall theorem).

Significance
The paper claims significance primarily in extending the classical addition-deletion philosophy from hyperplane arrangements to the context of plane curves. The main contribution is the derivation of explicit polynomial identities (Theorems 1.2 and 1.3) that allow for the computation of topological invariants (Poincaré and Betti polynomials) of a curve union based on its components.

As noted in Remark 3.6, the practical application of these results lies in the construction of new examples of free curves. By starting with known free arrangements and adding smooth or specific curves, one can compute the resulting Poincaré polynomial. If the polynomial splits into linear factors with integer roots, it suggests the freeness of the new arrangement, providing a pathway to verify freeness properties without direct, often difficult, algebraic computation of syzygies. The paper remains modest, presenting these identities as tools for understanding properties under the addition technique rather than claiming a complete classification of all plane curves.