[Paper Review] Torsion divisors of plane curves and Zariski pairs

Enrique Artal Bartolo, Shinzo Bannai+2 more|arXiv (Cornell University)|Oct 15, 2019

Algebraic Geometry and Number Theory8 references4 citations

TL;DR

This paper establishes a criterion for distinguishing the embedded topology of reducible plane curves with a smooth irreducible component using torsion classes in the degree-zero Picard group of that component. By constructing explicit examples via algebraic geometry and Galois covers, the authors prove the existence of a Zariski 4-tuple, demonstrating that torsion divisors serve as effective invariants beyond fundamental group and Alexander polynomials.

ABSTRACT

In this paper we study the embedded topology of reducible plane curves having a smooth irreducible component. In previous studies, the relation between the topology and certain torsion classes in the Picard group of degree zero of the smooth component was implicitly considered. We formulate this relation clearly and give a criterion for distinguishing the embedded topology in terms of torsion classes. Furthermore, we give a method of systematically constructing examples of curves where our criterion is applicable, and give new examples of Zariski tuples.

Motivation & Objective

• To clarify the implicit relationship between embedded topology and torsion classes in the Picard group of a smooth component in reducible plane curves.
• To develop a concrete criterion for distinguishing embedded topologies using torsion divisors.
• To systematically construct examples where the criterion applies, particularly focusing on curves with smooth components and multiple tangent lines or conics.
• To provide new examples of Zariski tuples, including a Zariski 4-tuple, by leveraging geometric group structures on elliptic curves.
• To demonstrate that torsion invariants can detect topological differences even when the fundamental group is abelian, thus offering a finer invariant than classical invariants.

Proposed method

• The authors use the geometric group structure of a smooth cubic curve to analyze the positions of tangent points and construct curves with prescribed intersection properties.
• They define and compute splitting invariants derived from Galois covers, particularly focusing on the behavior of components under pullback to covers.
• The construction relies on divisor linear equivalence on the smooth component, using the condition that a divisor is torsion if its multiple is linearly equivalent to zero.
• They apply the theory of plane curves of type $(d_0,d_1; n, k)$, where $k$ counts the number of conics tangent to the curve at $k$ points with multiplicity two.
• Explicit examples are built via algebraic manipulation of homogeneous polynomials defining curves, ensuring smoothness and correct intersection multiplicities.
• Computational verification is supported using Sagemath and Binder, with code available on GitHub for reproducibility.

Experimental results

Research questions

• RQ1 Can torsion classes in the Picard group of a smooth component detect differences in the embedded topology of plane curves with the same combinatorics?
• RQ2 How can one systematically construct plane curves where torsion invariants distinguish their embedded topologies?
• RQ3 What is the maximal number of conics that can be tangent to a fixed smooth curve at multiple points with multiplicity two, and how does this affect topology?
• RQ4 Can Zariski tuples of size four be constructed using torsion invariants when the fundamental group is abelian?
• RQ5 What role does the group law on an elliptic curve play in classifying the topological types of arrangements of curves tangent to it?

Key findings

• The paper proves the existence of a Zariski 4-tuple consisting of plane curves of type $(4,6;6,1)$, $(4,6;6,2)$, $(4,6;6,3)$, and $(4,6;6,6)$, all with the same combinatorics but different embedded topologies.
• A new criterion is established that distinguishes embedded topologies via the non-triviality of certain torsion classes in the Picard group of the smooth component.
• Examples of curves with $k=6$ tangent conics (type $(4,6;6,6)$) are explicitly constructed using nodal cubics with non-torsion points and rational maps.
• The construction shows that for a smooth quartic and a sextic, the number of conics tangent at multiple points with multiplicity two can be controlled via divisor conditions on the cubic component.
• The method confirms that torsion invariants are effective even when the fundamental group is abelian, thus providing a finer invariant than the fundamental group or Alexander polynomial.
• Computational verification via Sagemath confirms the correctness of the constructions, and code is publicly available on GitHub for replication.