Polar exploration of complex surface germs

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.

Inner rates of finite morphisms

“…As an application of the inner rates formula (Theorem 3.1), we will perform polar exploration using the inner rates expanding the ideas of [BdSFP22a,BdSFNP22,BdSFP22b]. The main aim of this section is to prove Proposition C .…”

“…As an application of our methods, we study the problem of polar exploration associated to Φ, following [BdSFP22a,BdSFNP22]. Roughly speaking, it is the study of the relative position on (X, 0) of the two curves Π Φ and (gf ) −1 (0) i.e., of the relative positions of their strict transform by a resolution of (X, 0).…”

“…More precisely it is the problem of determining the embedded topological type (Definition 9.1) of the union Π Φ ∪ (gf ) −1 (0) from that of the curve (gf ) −1 (0). This is a natural problem which has been studied by many authors such as Merle, García Barroso, Delgado, Maugendre, Kuo, Parusiński, Michel, Belotto, Fantini, Némethi, Pichon, among others (see, e.g.,[Mer77, KP04, GB00, DM03, Mic08, MM20, DM21, BdSFNP22,BdSFP22b]). An important contribution in the general case was made by Michel in [Mic08] via the Hironaka quotients of the morphism Φ (Definition 7.1).…”

“…An important contribution in the general case was made by Michel in [Mic08] via the Hironaka quotients of the morphism Φ (Definition 7.1). New techniques and results involving inner rates were recently developed in [BdSFNP22] in the case where Φ is a generic linear projection of (X, 0), and a complete answer to polar exploration was given in [BdSFP22b] in the case where (X, 0) is a Lipschitz normally embedded surface germ. In the present paper, we address the case of a general finite morphism Φ.…”