Trisections on Certain Rational Elliptic Surfaces and Families of Zariski Pairs Degenerating to the same Conic-line Arrangement

TL;DR

This paper explores the geometry of trisections on rational elliptic surfaces, using Mumford representations to explicitly construct plane curves and Zariski pairs, including a family degenerating to the same conic-line arrangement.

Contribution

It introduces a method to construct explicit trisections and Zariski pairs on rational elliptic surfaces, including a novel family degenerating to identical conic-line arrangements.

Findings

Constructed new examples of Zariski pairs.

Established a family of Zariski pairs degenerating to the same conic-line arrangement.

Utilized Mumford representations for explicit geometric constructions.

Abstract

In this paper, we study the geometry of trisections on certain rational elliptic surfaces. We utilize Mumford representations of semi-reduced divisors in order to construct trisections and related plane curves with interesting properties explicitly. As a result we are able to construct new examples of Zariski pairs. Especially, we show the existence of a family of Zariski pairs that degenerate to the same conic-line arrangement.