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Poncelet’s closure theorem and the embedded topology of conic-line arrangements
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 25 November 2024, pp. 1-15
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In the study of plane curves, one of the problems is to classify the embedded topology of plane curves in the complex projective plane that have a given fixed combinatorial type, where the combinatorial type of a plane curve is data equivalent to the embedded topology in its tubular neighborhood. A pair of plane curves with the same combinatorial type but distinct embedded topology is called a Zariski pair. In this paper, we consider Zariski pairs consisting of conic-line arrangements that arise from Poncelet’s closure theorem. We study unramified double covers of the union of two conics that are induced by a
$2m$-sided Poncelet transverse. As an application, we show the existence of families of Zariski pairs of degree 
$2m+6$ for 
$m\geq 2$ that consist of reducible curves having two conics and 
$2m+2$ lines as irreducible components.
