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arXiv · 2103.15444

Polar exploration of complex surface germs

André Belotto da Silva·Lorenzo Fantini·András Némethi·Anne Pichon
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Abstract

We prove that the topological type of a normal surface singularity $(X,0)$ provides finite bounds for the multiplicity and polar multiplicity of $(X,0)$, as well as for the combinatorics of the families of generic hyperplane sections and of polar curves of the generic plane projections of $(X,0)$. A key ingredient in our proof is a topological bound of the growth of the Mather discrepancies of $(X,0)$, which allows us to bound the number of point blowups necessary to achieve factorization of any resolution of $(X,0)$ through its Nash transform. This fits in the program of polar explorations, the quest to determine the generic polar variety of a singular surface germ, to which the final part of the paper is devoted.

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André Belotto da Silva, Lorenzo Fantini, András Némethi, Anne Pichon. 2021-03-29. Polar exploration of complex surface germs. https://doi.org/10.1090/tran%2F8749

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