POLAR CREMONA TRANSFORMATIONS AND MONODROMY OF POLYNOMIALS

Consider the gradient map associated to any non-constant homogeneous polynomial f is an element of C [x(0),..., x(n)] of degree d, defined by phi(f) = grad (f) : D(f) -> P-n, (x(0) : ... : x(n)) -> (f(0)(x) : ...: f(n)(x)) where D(f) = {x is an element of P-n; f (x) not equal 0} is the principal open set associated to f and f(i) = partial derivative f/partial derivative x(i). This map corresponds to polar Cremona transformations. In Proposition 3.4 we give a new lower bound for the degree d (f) of phi(f) 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.