On $\mathscr{M}$-arrangements of conics and lines with ordinary singularities

Technical Summary: On M-arrangements of Conics and Lines with Ordinary Singularities

Problem Statement and Context
This paper investigates the combinatorial and homological properties of a specific class of reduced plane curves known as M-curves. Introduced recently by Janasz and Pokora [10] as a generalization of classical "maximizing curves" (Persson [17]), M-curves are defined by having only ADE-singularities and simple elliptic singularities (ordinary quadruple points and $J_{2,0}$-singularities) while satisfying a specific numerical condition on the degree of their Jacobian ideal.

While maximizing curves of even degree are well-studied, those of odd degree are rare and difficult to construct. The authors aim to explore a slightly broader class of curves that retain the key numerical and homological properties of maximizing curves. The specific focus of this work is M-arrangements of conics and lines ($CL$) that admit only ordinary singularities of multiplicity at most four (double, triple, and quadruple points). This setting is motivated by the restrictive nature of M-curves (e.g., real M-arrangements of lines are necessarily simplicial [11]) and their connection to algebraic surfaces with extremal properties [15, 18, 19].

Methodology
The authors employ a combination of combinatorial counting, algebraic geometry, and syzygy theory:

1. Combinatorial Constraints: Using Bézout's Theorem and the definition of the total Tjurina number $\tau(C)$, the authors derive linear equations relating the number of lines ($d$), conics ($k$), and the counts of singular points of multiplicity $r$ ($n_r$).
2. Syzygy Theory and Freeness: The study relies heavily on the concept of free curves, where the Jacobian ideal is saturated. The authors utilize the minimal degree of Jacobian syzygies, denoted $mdr(f)$, and the exponents $(d_1, d_2)$ of free arrangements.
3. Poincaré Polynomials: The authors analyze the combinatorial Poincaré polynomial $P(CL; t)$, which for free arrangements factors into $(1+d_1t)(1+d_2t)$. They investigate how this polynomial behaves when a smooth conic is deleted from an M-arrangement.
4. Real Geometry Bounds: For real arrangements, the authors apply Melchior's inequality to derive boundedness results for the degree of such arrangements.
5. Regularity Calculations: The paper computes the Castelnuovo–Mumford regularity of the Milnor algebra $M(f)$ and the module of Jacobian syzygies $AR(f)$ using relations between stability thresholds, coincidence thresholds, and the exponents of free curves.

Key Contributions and Results

• Numerical Constraints on Weak Combinatorics (Theorem 3.1):
  The authors establish necessary conditions for the weak combinatorics of an M-arrangement consisting of $k$ conics and $d$ lines with ordinary singularities of multiplicity $\le 4$.

  • If $d = 2\ell$ (even), the counts must satisfy: $n_2 + 2n_3 + 3n_4 = (k + \ell)^2 + \ell - k - 3$.
  • If $d = 2\ell + 1$ (odd), the counts must satisfy: $n_2 + 2n_3 + 3n_4 = (k + \ell)^2 + 2\ell - 1$.
    These constraints allow for the elimination of non-existent combinatorial types (Example 3.2).

• Specialization to One Conic and $d$ Lines (Proposition 3.4 & 3.6):
  Focusing on the case $k=1$, the authors derive explicit systems of equations for $n_2, n_3, n_4$. Furthermore, they provide strict constraints on the number of intersection points $r$ between the conic and the remaining lines, showing that $r$ cannot take arbitrary values but is bounded by specific functions of $\ell$.

• Combinatorial Poincaré Polynomials (Theorem 4.3):
  The paper provides an explicit formula for the Poincaré polynomial of the line arrangement $L$ obtained by deleting the conic from an M-arrangement $CL$.

  • For $d=2\ell$, $P(L; t) = (\ell^2 + \ell - 2 - r)t^2 + (2\ell - 1)t + 1$.
  • For $d=2\ell+1$, $P(L; t) = (\ell^2 + 2\ell - r)t^2 + 2\ell t + 1$.
    The authors demonstrate that while the deletion of a conic from an M-arrangement often yields a free line arrangement (e.g., Examples 4.5, 4.6, 4.8), it does not guarantee freeness (Example 4.7), as the polynomial may fail to split over the rationals.

• New Construction (Example 4.8):
  The authors construct a new explicit example of an M-arrangement consisting of one conic and eleven lines. This arrangement has weak combinatorics $C(11, 1; 11, 2, 10)$, total Tjurina number $\tau = 109$, and exponents $(5, 7)$. The deletion of the conic results in a free arrangement of 11 lines with exponents $(5, 5)$.

• Boundedness of Real Arrangements (Theorem 5.1):
  The paper proves that real M-arrangements of one conic and lines are bounded in degree.

  • If $d = 2\ell$, then $\ell \le 9$ (no such arrangements for even $d \ge 20$).
  • If $d = 2\ell + 1$, then $\ell \le 10$ (no such arrangements for odd $d \ge 23$).

• Regularity of Associated Modules (Proposition 6.4):
  The authors record precise values for the Castelnuovo–Mumford regularity of the Milnor algebra $M(f)$ and the syzygy module $AR(f)$ for M-curves of degree $d$:

  • $\text{reg}(M(f)) = 3m - 2$ (if $d=2m$) or $3m - 1$ (if $d=2m+1$).
  • $\text{reg}(AR(f)) = m + 1$.

Significance and Claims
The paper claims to provide a "complete set of constraints" for the weak combinatorics of M-arrangements with a single conic, thereby characterizing the possible combinatorial types of such curves. It highlights the restrictive nature of M-curves, noting that they form a "delicate class" where numerical conditions tightly constrain geometric possibilities.

The authors position their work as complementary to existing classifications of arrangements with linear syzygies (Beorchia and Miró-Roig [2]) or those invariant under differential equations (de Moura Canaan and Coutinho [3]). While those works often involve non-ordinary singularities, this paper focuses strictly on ordinary singularities of low multiplicity. The construction of the new 11-line example and the proof of boundedness for real arrangements are presented as concrete advancements in understanding the existence and limits of these curves. The regularity results are framed as a contribution to the broader understanding of the homological properties of M-curves, linking them to the stability and coincidence thresholds of free curves.