On the polar degree of projective hypersurfaces

Abstract

Given a hypersurface in the complex projective $n$-space we prove several known formulas for the degree of its polar map by purely algebro-geometric methods. Furthermore, we give formulas for the degree of its polar map in terms of the degrees of the polar maps of its components. As an application, we classify the plane curves with polar map of low degree, including a very simple proof of I. Dolgachev's classification of homaloidal plane curves.

11 pages, 2 figures. Fixed minor typos. Final version, to appear in J. London Math. Soc

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• Polar degree of hypersurfaces with 1-dimensional singularities
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• Polar degree and vanishing cycles