[Paper Review] A note on splitting numbers for Galois covers and $\pi_1$-equivalent Zariski $k$-plets

Taketo Shirane|arXiv (Cornell University)|Jan 15, 2016

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper introduces splitting numbers for subvarieties in smooth complex varieties under Galois covers, proving they are invariant under certain homeomorphisms. By applying this invariant to Shimada's equisingular families of plane curves of type (b, m), the authors show that curves in distinct connected components of the family are topologically distinguishable even when their fundamental groups are isomorphic. The key result is the construction of π₁-equivalent Zariski k-plets for any k ≥ 2, resolving an open problem on the existence of such k-plets beyond pairs.

ABSTRACT

In this paper, we introduce extit{splitting numbers} of subvarieties in a smooth variety for a Galois cover, and prove that the splitting numbers are invariant under certain homeomorphisms. By splitting numbers, we give a necessary and sufficient condition for two plane curves of type $(b,m)$ to be topologically equivalent as pairs of the complex projective plane and plane curves, where a plane curve of type $(b,m)$ is an arrangement of two smooth plane curves of degree $3$ and $b$ defined by I.~Shimada. Consequently, we prove that there are $\pi_1$-equivalent Zariski $k$-plets for any $k\geq2$.

Motivation & Objective

• To define and study splitting numbers of subvarieties under Galois covers as a topological invariant.
• To resolve Problem 0.3 on whether curves of type (b, m) in distinct connected components of Fb,m form a Zariski pair.
• To prove the existence of π₁-equivalent Zariski k-plets for any k ≥ 2, extending known results on Zariski pairs.

Proposed method

• Define the splitting number sφ(C) of a subvariety C under a Galois cover φ as the number of connected components of the pullback of C in the normalization of the cover.
• Use étale covers and the geometry of line bundles to relate splitting numbers to the order of divisors in Pic0 of the curve.
• Apply the theory of simple cyclic covers of degree m branched along a curve B to compute splitting numbers of curves C intersecting B with multiplicity divisible by m.
• Leverage the structure of Shimada’s equisingular families Fb,m(µ) to relate the splitting number of E to the order of a divisor class in Pic0(E).
• Prove that splitting numbers are invariant under homeomorphisms preserving the cover structure, using normalization and birational maps.
• Use Theorem 2.7 to compute splitting numbers via the existence of intermediate curves of specific degrees.

Experimental results

Research questions

• RQ1 Can splitting numbers distinguish topologies of plane curves even when their fundamental groups are isomorphic?
• RQ2 Are two curves of type (b, m) in distinct connected components of Fb,m topologically distinguishable via splitting numbers?
• RQ3 Does every integer k ≥ 2 admit a π₁-equivalent Zariski k-plet?
• RQ4 Is the splitting number of a curve E in a Galois cover determined by the order of a divisor class in Pic0(E)?
• RQ5 Can the invariant sφ(C) detect non-homeomorphic embeddings of curves in P² when fundamental groups are isomorphic?

Key findings

• The splitting number sφ(C) is invariant under homeomorphisms that preserve the Galois cover structure.
• For curves of type (b, m), the splitting number of E is sφ(E) = m/µ, where µ is the order of [OE(DR − nH)] in Pic0(E).
• Curves R1 and R2 of type (b, m) in distinct connected components of Fb,m form a Zariski pair if and only if they have different splitting numbers.
• The number of connected components of Fb,m is equal to the number of divisors of m, and each component corresponds to a fixed splitting number.
• For any k ≥ 2, there exists a π₁-equivalent Zariski k-plet, constructed from the family F5k−1,5k−1 with k distinct components.
• The construction confirms that π₁-equivalent Zariski k-plets exist for all k ≥ 2, answering an open question in the field.