Around the polar degree conjectures

pol(V)=2. Dolgachev (Michigan Math J, 2000) has initiated the study of Cremona polar transformations i.e. birational maps gradf:ℙn⇢ℙn defined by the gradient map of a homogeneous polynomial. He conjectured that the topological degree of gradf depends only on the projective zero locus V of f, so that it can be called polar degree of V, denoted pol(V). The hypersurfaces with pol(V)=1 are called homaloidal; Dolgachev classified the homaloidal plane curves. Dimca and Papadima (Annals of Math 2003) conjectured classification of homaloidal hypersurfaces with isolated singularities was proved by Huh (Duke Math J, 2014). Huh conjectured  the classification of hypersurfaces with isolated singularities and pol(V)=2. We formulate in a precise way Huh's conjecture and give its proof.

f:ℙn−−>ℙn