摘要翻译：
考虑与任意一个非常齐次多项式$f\in\c[x_0,...,x_n]$相关的梯度映射，该多项式由\[\phi_f=grad(f):D(f)\to\cp^n,(x_0:...:x_n)\to(f_0(x):...:f_n(x))\]定义，其中$D(f)=\{x\in\cp^n；f(x)\neq0}$是与$f$和$f_i=\frac{\部分f}{\部分x_i}$相关的主开集。这张地图对应于极克雷莫纳变换。在命题\ref{p1}中，我们在射影超曲面$v:f=0$只有孤立奇点的假设下，给出了$\phi_f$的度$d(f)$的一个新的下界。当$d(f)=1$时，定理{t4}对$v$的奇性给出了很强的条件。
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英文标题：
《Polar Cremona Transformations and Monodromy of Polynomials》
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作者：
Imran Ahmed
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  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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PDF链接：
https://arxiv.org/pdf/0705.0709