Affiliation:
1. 1 Government College University School of Mathematical Sciences 68-B New Muslim Town Lahore Pakistan
Abstract
Consider the gradient map associated to any non-constant homogeneous polynomial
f
∈ ℂ[
x0
, ...,
xn
] of degree
d
, defined by \documentclass{aastex}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{mathrsfs}
\usepackage{pifont}
\usepackage{stmaryrd}
\usepackage{textcomp}
\usepackage{upgreek}
\usepackage{portland,xspace}
\usepackage{amsmath,amsxtra}
\usepackage{bbm}
\pagestyle{empty}
\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
$$\varphi _f = grad (f):D(f) \to \mathbb{P}^n ,(x_0 : \ldots :x_n ) \to (f_0 (x): \ldots :f_n (x))$$
\end{document} where
D
(
f
) = {
x
∈ ℙ
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 3.4 we give a new lower bound for the degree
d
(
f
) of ϕ
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
.
Reference15 articles.
1. On the classi-cation of cubic surfaces;Bruce J. W.;J. London Math. Soc.,1979
2. Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian;Ciliberto C.;Adv. Math.,2008
3. Complex Hypersurfaces Di-eomorphic to A-ne Spaces;Choudary A. D. R.;Kodai Math. J.,1994
4. Dimca, A. , Topics on Real and Complex Singularities, Vieweg, 1987. Adv. Lect. in Math. MR 92d :32048
5. Singularities and Topology of Hypersurfaces
Cited by
4 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献