Polar Cremona Transformations and Monodromy of Polynomials

Abstract

Consider the gradient map associated to any non-constant homogeneous polynomial f∈ [x0,...,xn] of degree d, defined by \[φf=grad(f): D(f) n, (x0:...:xn) (f0(x):...:fn(x))\] where D(f)=\x∈ n; f(x)≠ 0\ is the principal open set associated to f and fi=∂ f∂ xi. This map corresponds to polar Cremona transformations. In Proposition p1 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 t4 yields very strong conditions on the singularities of V.