Boundary of the Milnor fiber of a non-isolated Newton non degenerate surface singularity

Octave Curmi
Institut de Mathématiques Alfréd Rényi, Budapest
https://octavecurmi.github.io/index-fr.html

Date(s) : 18/03/2021   iCal
14 h 00 min - 15 h 00 min

Milnor fibers play a crucial role in the study of the topology of a singularity of surface. They correspond to the different possible smoothings of the singularity. A description of the full fiber is known in some particular cases, but not in general, even for isolated singularities. However, the study of its boundary has been an active field of research in the last decades. In different settings, this boundary has been proven to be a graph manifold. (Mumford, 1961, for isolated singularities, Michel-Pichon, 2003, 2014, for a smoothing of a reduced surface with smooth total space, Némethi-Szilard, 2012, with the same hypothesis, Bobadilla-Menegon Neto, 2014, for a non-reduced surface and a total space with isolated singularity).

The constructive proof provided by Némethi and Szilard can be adapted to prove the same result for a smoothing of a reduced surface with any total space. In the case when the total space is toric and the function giving the smoothing is Newton-non-degenerate, we provide a combinatorial algorithm for the computation of the boundary of the Milnor fiber.

I will expose the main lines of this algorithm and some of the natural questions which it leads to.

https://www.sciencedirect.com/science/article/abs/pii/S0001870820303078

https://arxiv.org/abs/1911.13258

 

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