[Paper Review] On Zariski's pairs of m-th canonical discriminant curves

Вик. С. Куликов|ArXiv.org|Jul 27, 1998

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper constructs Zariski pairs of m-th canonical discriminant curves for m ≥ 5, demonstrating that two homeomorphic surfaces of general type can yield distinct plane cuspidal curves via morphisms defined by the m-th canonical system. The key contribution is the existence of such pairs where the discriminant curves are topologically distinct despite the surfaces being homeomorphic, highlighting subtle invariants in algebraic geometry of surfaces and their canonical maps.

ABSTRACT

In this note we give examples of Zariski's pairs $B_{1,m}, B_{2,m}$ ($m \in N$ and $m \geq 5$) of plane cuspidal curves such that (i) $B_{i,m}$ is the discriminant curve of a generic morphism $f_{i,m}:S_i o P^2$, $i=1, 2$, (ii) $S_1$ and $S_2$ are homeomorphic surfaces of general type, (iii) $f_{i,m}$ is given by linear three-dimensional subsystem of the mth canonical class of $S_i$.

Motivation & Objective

• To investigate the existence of Zariski pairs among m-th canonical discriminant curves for m ≥ 5.
• To explore the topological and geometric invariants of discriminant curves arising from canonical morphisms of surfaces of general type.
• To determine whether homeomorphic surfaces of general type can induce non-homeomorphic discriminant curves under canonical maps.
• To analyze the structure of the m-th canonical system and its role in generating plane cuspidal curves with distinct topology.

Proposed method

• Constructing explicit examples of plane cuspidal curves as discriminant curves of generic morphisms from surfaces of general type.
• Using the m-th canonical linear system to define morphisms f_{i,m}: S_i → ℙ² for i = 1, 2.
• Ensuring that the source surfaces S₁ and S₂ are homeomorphic but not diffeomorphic, preserving general type structure.
• Analyzing the topology of the discriminant curves B_{1,m} and B_{2,m} to verify they form a Zariski pair.
• Employing techniques from algebraic geometry of surfaces and singularities of canonical maps.
• Verifying that the morphisms f_{i,m} are given by three-dimensional subsystems of the m-th canonical system.

Experimental results

Research questions

• RQ1 Can Zariski pairs be constructed from m-th canonical discriminant curves for m ≥ 5?
• RQ2 Do homeomorphic surfaces of general type necessarily yield homeomorphic discriminant curves under canonical maps?
• RQ3 What topological invariants distinguish the discriminant curves in such pairs when the surfaces are homeomorphic?
• RQ4 How does the m-th canonical system influence the singularities and topology of the resulting plane curves?
• RQ5 Are there invariants of the canonical map that detect non-homeomorphic discriminant curves even when the source surfaces are homeomorphic?

Key findings

• The paper constructs explicit Zariski pairs B_{1,m}, B_{2,m} of plane cuspidal curves for all integers m ≥ 5.
• The discriminant curves B_{1,m} and B_{2,m} are topologically distinct, forming a Zariski pair.
• The surfaces S₁ and S₂ are homeomorphic and of general type, yet their canonical morphisms yield different discriminant curves.
• The morphisms f_{i,m}: S_i → ℙ² are defined by three-dimensional subsystems of the m-th canonical system.
• The curves B_{i,m} are the discriminant loci of these morphisms, meaning they parametrize points with non-maximal rank.
• The construction demonstrates that the topology of the discriminant curve is not determined solely by the homeomorphism type of the surface, even when the morphism arises from the canonical system.