Polar Cremona Transformations and Monodromy of Polynomials

Abstract

Consider the gradient map associated to any non-constant homogeneous polynomial $f\in \C[x_0,...,x_n]$ of degree $d$, defined by \[ϕ_f=grad(f): D(f)\to \CP^n, (x_0:...:x_n)\to (f_0(x):...:f_n(x))\] where $D(f)=\{x\in \CP^n; f(x)\neq 0\}$ is the principal open set associated to $f$ and $f_i=\frac{\partial f}{\partial x_i}$. This map corresponds to polar Cremona transformations. In Proposition \ref{p1} we give a new lower bound for the degree $d(f)$ of ${\varphi }_f$ under the assumption that the projective hypersurface $V:f=0$ has only isolated singularities. When $d(f)=1$, Theorem \ref{t4} yields very strong conditions on the singularities of $V$.

8 pages

References in corpus (2)

• Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian
• On the degree of Polar Transformations -- An approach through Logarithmic Foliations

Cited by in corpus (2)

• Euler characteristics of general linear sections and polynomial Chern classes
• Milnor numbers of projective hypersurfaces with isolated singularities