Polar exploration of complex surface germs

Belotto da Silva, André; Fantini, Lorenzo; Némethi, András; Pichon, Anne

doi:10.1090/tran/8749

Polar exploration of complex surface germs
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by André Belotto da Silva, Lorenzo Fantini, András Némethi and Anne Pichon PDF

Trans. Amer. Math. Soc. 375 (2022), 6747-6767

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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