This research was supported by the Narodowe Centrum Nauki under the OPUS grant number 2023/49/B/ST1/00848.
Title of the project - On Zariski pairs of surface singularities
Project's period - 15.07.2024 - 14.07.2026
Team - Christophe Eyral (PI), Masaharu Ishikawa, Mutsuo Oka, Öznur Turhan
Many mathematical objects develop singularities—points where regularity breaks down. The study of such phenomena, known as singularity theory, aims to understand the structure of these irregularities and to classify the various forms they may exhibit. Understanding when two singularities should be considered the same, and when subtle differences remain hidden behind identical numerical characteristics, is one of the fundamental questions in this field.
Within this broad context, the project focused on a particularly challenging problem concerning μ-Zariski pairs. These are pairs of isolated surface singularities that share many of their main mathematical properties and invariants but nevertheless cannot be continuously deformed into one another by deformations preserving these invariants, in particular their Milnor number μ. Such examples are extremely valuable because they reveal that the classical tools used to classify singularities do not always capture their full geometric complexity.
The first examples of such, and even stronger, phenomena were constructed in the 1990s by Artal Bartolo. A key achievement of the present project is the development of a general framework for producing new examples of μ-Zariski pairs. This framework extends previous constructions from classical Lê–Yomdin singularities to weighted–Lê–Yomdin singularities. Beyond this extension, the project establishes that the phenomenon also occurs outside these families, significantly broadening the scope of known examples and opening new directions for future research.
The project also produced an additional result that was not originally included among its objectives. The preliminary investigation of resolution techniques arising naturally in the study of μ-Zariski pairs led to an independent result concerning simultaneous resolutions of certain families of surface singularities. This result also yielded a new geometric proof of a weak version of a classical theorem of Milnor and Orlik concerning the monodromy zeta function and, in particular, the Milnor number of weighted homogeneous isolated singularities.
Overall, the project successfully achieved all of its planned objectives and significantly advanced the understanding of complex surface singularities. It provides a new framework for analysing subtle phenomena involving such singularities, clarifies the limitations of classical invariants, and in doing so strengthens the connections between algebraic geometry, topology, and deformation theory. While this research is fundamental rather than directly application-oriented, advances of this kind expand the mathematical foundations on which many other areas of mathematics are built and deepen our understanding of the geometry of singular spaces.
Publications
C. Eyral and M. Oka, On Milnor-Orlik's theorem and admissible simultaneous good resolutions, Ann. Polon. Math. 134 (2025), no. 2, 107-118.
C. Eyral, M. Ishikawa, and M. Oka, Newton weighted–Lê–Yomdin polynomials and µ-Zariski pairs of surface singularities, arXiv:2511.06939, submitted for publication
C. Eyral, M. Ishikawa, M. Oka, and Ö. Turhan, New µ-Zariski pairs of surface singularities, arXiv:2604.03018, submitted for publication