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On arrangements of smooth plane quartics and their bitangents

Marek Janasz, Piotr Pokora, Marcin Zieliński

TL;DR

The work investigates the geometry of smooth plane quartics and their bitangents, focusing on weak combinatorics and freeness in quartic–line arrangements with quasi-homogeneous singularities. It develops a Hirzebruch-type inequality to constrain quadruple intersections, and analyzes specific highly symmetric quartics (Klein, Dyck, Komiya–Kuribayashi) to reveal incidence structures among their 28 bitangents and ties these to automorphisms and bielliptic involutions. The authors also construct new 3-syzygy (plus-one generated) plane curves by combining quartics with lines, providing explicit examples and homological data computed via $\text{SINGULAR}$. These results contribute to both combinatorial geometry of bitangents and the algebraic study of Milnor algebras, extending the understanding of freeness and syzygy phenomena in curve arrangements.

Abstract

In the present paper, we revisit the geometry of smooth plane quartics and their bitangents from several perspectives. First, we study in detail the weak combinatorics of arrangements of bitangents associated with highly symmetric quartic curves. We consider quartic curves from the point of view of the order of their automorphism groups, in order to establish a lower bound on the number of quadruple intersection points for arrangements of bitangents associated with smooth plane quartics, which are smooth members of Ciani's pencil. We then construct new examples of $3$-syzygy reduced plane curves using smooth plane quartics and their bitangents.

On arrangements of smooth plane quartics and their bitangents

TL;DR

The work investigates the geometry of smooth plane quartics and their bitangents, focusing on weak combinatorics and freeness in quartic–line arrangements with quasi-homogeneous singularities. It develops a Hirzebruch-type inequality to constrain quadruple intersections, and analyzes specific highly symmetric quartics (Klein, Dyck, Komiya–Kuribayashi) to reveal incidence structures among their 28 bitangents and ties these to automorphisms and bielliptic involutions. The authors also construct new 3-syzygy (plus-one generated) plane curves by combining quartics with lines, providing explicit examples and homological data computed via . These results contribute to both combinatorial geometry of bitangents and the algebraic study of Milnor algebras, extending the understanding of freeness and syzygy phenomena in curve arrangements.

Abstract

In the present paper, we revisit the geometry of smooth plane quartics and their bitangents from several perspectives. First, we study in detail the weak combinatorics of arrangements of bitangents associated with highly symmetric quartic curves. We consider quartic curves from the point of view of the order of their automorphism groups, in order to establish a lower bound on the number of quadruple intersection points for arrangements of bitangents associated with smooth plane quartics, which are smooth members of Ciani's pencil. We then construct new examples of -syzygy reduced plane curves using smooth plane quartics and their bitangents.
Paper Structure (7 sections, 19 theorems, 28 equations, 10 tables)

This paper contains 7 sections, 19 theorems, 28 equations, 10 tables.

Table of Contents

  1. Introduction
  2. Hirzebuch-type inequality for arrangements of smooth plane quartics and lines with quasi-homogeneous singularities
  3. Smooth plane quartics and their bitangents
  4. The Klein quartic curve and its bitangents
  5. The Fermat quartic curve and its bitangents
  6. The Komiya--Kuribayashi quartic curve and its bitangents
  7. 3-syzygy curves constructed with plane quartics and lines

Key Result

Theorem 2.1

Let $\mathcal{QL} = \{\ell_{1}, \ldots , \ell_{d}, Q_{1}, \ldots , Q_{k}\} \subset \mathbb{P}^{2}_{\mathbb{C}}$ be an arrangement of $d\geqslant 1$ lines and $k\geqslant 1$ smooth quartics such that $4k+d\geqslant 6$. Assume that $\mathcal{QL}$ admits $n_{2}$ nodes, $t_{2}$ tacnodes, $t_{5}$ singula

Theorems & Definitions (38)

  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 28 more