Polar exploration of complex surface germs

Belotto da Silva, André and Fantini, Lorenzo and Némethi, András and Pichon, Anne (2022) Polar exploration of complex surface germs. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 375 (9). pp. 6747-6767. ISSN 0002-9947

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Abstract

We prove that the topological type of a normal surface singularity ( X , 0 ) (X,0) provides finite bounds for the multiplicity and polar multiplicity of ( X , 0 ) (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 ) (X,0) . A key ingredient in our proof is a topological bound of the growth of the Mather discrepancies of ( X , 0 ) (X,0) , which allows us to bound the number of point blowups necessary to achieve factorization of any resolution of ( X , 0 ) (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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