Polar Cremona Transformations and Monodromy of Polynomials

Consider the gradient map associated to any non-constant homogeneous polynomial f ∈ C[x0, . . . , xn] of degree d, defined by φf = grad(f) : D(f) → P , (x0 : . . . : xn) → (f0(x) : . . . : fn(x)) whereD(f) = {x ∈ P; f(x) 6= 0} is the principal open set associated to f and fi = ∂f ∂xi . This map corresponds to polar Cremona transformations. In Proposition 3.4 we give a new lower bound for the degree d(f) of φf under the assumption that the projective hypersurface V : f = 0 has only isolated singularities. When d(f) = 1, Theorem 4.2 yields very strong conditions on the singularities of V .