Polar Cremona Transformations and Monodromy of Polynomials

TL;DR

This paper investigates the properties of gradient maps associated with homogeneous polynomials, providing new bounds on their degree and conditions on singularities when the degree is minimal, within the context of polar Cremona transformations.

Contribution

It introduces a new lower bound for the degree of gradient maps of homogeneous polynomials with isolated singularities and characterizes singularities when the degree equals one.

Findings

New lower bound for the degree of gradient maps

Strong conditions on singularities when degree is one

Insights into polar Cremona transformations

Abstract

Consider the gradient map associated to any non-constant homogeneous polynomial $f\in \C[x_0,...,x_n]$ of degree $d$, defined by \[\phi_f=grad(f): D(f)\to \CP^n, (x_0:...:x_n)\to (f_0(x):...:f_n(x))\] 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 $\phi_f$ 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 $V$.