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

Marek Janasz, Piotr Pokora

TL;DR

The paper analyzes M-arrangements of conics and lines with ordinary singularities, establishing parity-dependent relations between degree and Jacobian data to extend the theory of maximizing curves. It develops a weak combinatorics framework and a combinatorial criterion for the existence of such arrangements, then provides explicit formulas for the combinatorial Poincaré polynomials of line arrangements obtained by deleting a single conic, including detailed examples that display freeness and plus-one generated phenomena. It further examines obstructions and exact Poincaré polynomials in the deletion scenario, and closes with a precise computation of Castelnuovo-Mumford regularity for the associated Jacobian modules, linking geometric constraints to homological invariants. Overall, the work integrates combinatorial, algebraic, and homological perspectives to illuminate the structure of M-arrangements and their role in surface constructions.

Abstract

In this paper, we study combinatorial aspects of reduced plane curves known as $\mathscr{M}$-curves. This notation is a natural generalization of maximizing plane curves which are well-known in the theory of algebraic surfaces. We focus here on $\mathscr{M}$-arrangements of conics and lines with ordinary singularities of multiplicity less than five and we provide various numerical constraints on their existence, particularly in terms of their weak combinatorics. Moreover, we study in detail the scenario when our $\mathscr{M}$-arrangements consist of lines and just one conic.

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

TL;DR

The paper analyzes M-arrangements of conics and lines with ordinary singularities, establishing parity-dependent relations between degree and Jacobian data to extend the theory of maximizing curves. It develops a weak combinatorics framework and a combinatorial criterion for the existence of such arrangements, then provides explicit formulas for the combinatorial Poincaré polynomials of line arrangements obtained by deleting a single conic, including detailed examples that display freeness and plus-one generated phenomena. It further examines obstructions and exact Poincaré polynomials in the deletion scenario, and closes with a precise computation of Castelnuovo-Mumford regularity for the associated Jacobian modules, linking geometric constraints to homological invariants. Overall, the work integrates combinatorial, algebraic, and homological perspectives to illuminate the structure of M-arrangements and their role in surface constructions.

Abstract

In this paper, we study combinatorial aspects of reduced plane curves known as -curves. This notation is a natural generalization of maximizing plane curves which are well-known in the theory of algebraic surfaces. We focus here on -arrangements of conics and lines with ordinary singularities of multiplicity less than five and we provide various numerical constraints on their existence, particularly in terms of their weak combinatorics. Moreover, we study in detail the scenario when our -arrangements consist of lines and just one conic.
Paper Structure (5 sections, 7 theorems, 47 equations, 3 figures)

This paper contains 5 sections, 7 theorems, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constraints on weak-combinatorics of $\mathscr{M}$-arrangements of conics and lines
  4. Combinatorial Poincaré polynomials for $\mathscr{M}$-arrangements of conics and lines
  5. A remark on the Castelnuovo-Mumford regularity of $\mathscr{M}$-curves

Key Result

Theorem 2.6

Let $C = \{f=0\}\subset \mathbb{P}^{2}_{\mathbb{C}}$ be a reduced curve of degree $d$ and let $r = {\rm mdr}(f)$. Let us denote the total Tjurina number of $C$ by $\tau(C)$. Then the following two cases hold.

Figures (3)

  • Figure 1: Arrangement $\mathcal{CL}_1$.
  • Figure 2: Arrangement $\mathcal{CL}_{2}$.
  • Figure 3: Arrangement $\mathcal{CL}_{3}$.

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6: du-Plessis -- Wall
  • Remark 2.7
  • Remark 2.8
  • Theorem 3.1
  • ...and 22 more