Polar exploration of complex surface germs

André Belotto da Silva , Lorenzo Fantini , András Némethi , Anne Pichon (1)

1 I2M - Institut de Mathématiques de Marseille

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.

Domaines: Géométrie algébrique [math.AG]

André Belotto da Silva, Lorenzo Fantini, András Némethi, Anne Pichon. Polar exploration of complex surface germs. Transactions of the American Mathematical Society, In press. ⟨hal-03580963⟩