Abstract.
We investigate the degree of the polar transformations associated to a certain class of multi-valued homogeneous functions. In particular we prove that the degree of the preimage of generic linear spaces by a polar transformation associated to a homogeneous polynomial F is determined by the zero locus of F. For zero dimensional-dimensional linear spaces this was conjectured by Dolgachev and proved by Dimca–Papadima using topological arguments. Our methods are algebro-geometric and rely on the study of the Gauss map of naturally associated logarithmic foliations.
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Fassarella, T., Pereira, J.V. On the degree of polar transformations. An approach through logarithmic foliations. Sel. math., New ser. 13, 239 (2007). https://doi.org/10.1007/s00029-007-0040-x
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DOI: https://doi.org/10.1007/s00029-007-0040-x