Classification of rational unicuspidal curves with two Newton pairs

TL;DR

This paper provides a comprehensive classification of the local topological types of singularities with two Newton pairs on rational unicuspidal complex projective plane curves, confirming realizability for most types.

Contribution

It completes the classification of such singularities and verifies the realizability of all but two types, building on Liu's thesis and prior work by several researchers.

Findings

Most local types are realizable by existing constructions.

Two local types remain non-realizable.

The classification advances understanding of singularities on rational unicuspidal curves.

Abstract

Based on Tiankai Liu's PhD thesis (MIT, 2014), we give a complete classification of local topological types of singularities with two Newton pairs on rational unicuspidal complex projective plane curves. We show that all but two possible local types on Liu's list are realizable by the work of H. Kashiwara, M. Miyanishi, T. Sugie, T. Fenske, K. Tono and S. Y. Orevkov.