Polar Cremona Transformations and Monodromy of Polynomials

Details Polar Cremona Transformations and Monodromy of Polynomials Polar Cremona Transformations and Monodromy of Polynomials

2007-05-05

arxiv.org/abs/0705.0709v1

Studia Scientiarum Mathematicarum Hungarica 47(1), 81-89 (2010)

Mathematics

Algebraic Geometry

8 pages

Scientific paper

10.1556/SScMath.2009.1114

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 $V$.

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