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edited Oct 9 2010 at 11:47 |
enter code here -
The simplest example of a singular
algebraic variety which is a
topological manifold is given by the
cusp $$z_1^2-z_0^3=0.$$ The cusp is a topological manifold
homeomorphic to a real plane
$\mathbb{R}^2$ as can be seen by
the parametrization $t\mapsto (z_1, z_0)= z_1,z_0)= (t^2,t^3)$ where $t$ is a complex parametervariable. Mumford has proven that a two
dimensional normal complex space
which is a topological manifold is
always nonsingular. Mumford's result does not generalize
to (odd) dimensions higher than 2 as
proven by Brieskorn using the
following counter examples which generalizes the case of the cusp:
$$z_1^2+ z_2^2+\cdots
z_{2k+1}^2-z_0^3=0,\quad \text{where} \quad k\in
\mathbb{N}_0.$$ More generally, given $a=(a_1,
\cdots, a_n)\in \mathbb{N}^n_0$ with $a_j>1$ for all $j$, one
can define the following variety
$\Gamma(a)$ known as a
Brieskorn-Pham variety: $$ \Gamma(a): \quad z_1^{a_1}+\cdots
z_n^{a_n}=0. $$ - Brieskorn has proved the following conjecture of Milnor:
$$\Gamma(a)\quad \text{is a topological manifold}
\iff \prod_{1\leq k_l\leq
a_k-1}(1-\epsilon_1^{k_1}
\epsilon_1^{k_2}\cdots
\epsilon_n^{k_n} )=1,$$ where
$\epsilon_k=\mathrm{exp}\Big({\frac{2\pi
}{a_k}\mathrm{i} }\Big)$ for $k=1,\cdots, n$.
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edited Oct 8 2010 at 20:11 |
enter code here - The simplest example of a singular
algebraic variety which is a
topological manifold is given by the
cusp
$$z_1^2-z_0^3=0.$$
The cusp is
a topological manifold homeomorphic
to a real plane $\mathbb{R}^2$ as can
be seen by the parametrization $t\mapsto (z_1, z_0)= (t^2,t^3)$ where $t$ is a
complex parameter.
Mumford has proven that a two
dimensional normal complex space
which is a topological manifold is
always nonsingular. Munford's Mumford's result does not generalize
to (odd) dimensions higher than 2 as
proven by Brieskorn using the
following counter examples which generalizes the case of the cusp:
$$z_1^2+ z_2^2+\cdots
z_{2k+1}^2-z_0^3=0,\quad \text{where} \quad k\in
\mathbb{N}_0.$$ More generally, given $a=(a_1,
\cdots, a_n)\in \mathbb{N}^n_0$ with $a_j>1$ for all $j$, one
can define the following variety
$\Gamma(a)$ known as a
Brieskorn-Pham variety: $$ \Gamma(a): \quad z_1^{a_1}+\cdots
z_n^{a_n}=0. $$ - Brieskorn has proved the following conjecture of Milnor:
$$\Gamma(a)\quad \text{is a topological manifold}
\iff \prod_{1\leq k_l\leq
a_k-1}(1-\epsilon_1^{k_1}
\epsilon_1^{k_2}\cdots
\epsilon_n^{k_n} )=1,$$ where
$\epsilon_k=\mathrm{exp}\Big({\frac{2\pi
}{a_k}\mathrm{i} }\Big)$ for $k=1,\cdots, n$.
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edited Oct 8 2010 at 17:30 |
enter code here - The simplest example of a singular
algebraic variety which is a
topological manifold is given by the
cusp
$$z_1^2-z_0^3=0.$$
The cusp is
a topological manifold homeomorphic
to a real plane $\mathbb{R}^2$ as can
be seen by the parametrization $t\mapsto (z_1, z_0)= (t^2,t^3)$ where $t$ is a
complex parameter.
Munford
Mumford has proven that a two
dimensional normal complex space
which is a topological manifold is
always nonsingular. Munford's result does not generalize
to (odd) dimensions higher than 2 as
proven by Brieskorn using the
following counter examples which generalizes the case of the cusp:
$$z_1^2+ z_2^2+\cdots
z_{2k+1}^2-z_0^3=0,\quad \text{where} \quad k\in
\mathbb{N}_0.$$ More generally, given $a=(a_1,
\cdots, a_n)\in \mathbb{N}^n_0$ with $a_j>1$ for all $j$, one
can define the following variety
$\Gamma(a)$ known as a
Brieskorn-Pham variety: $$ \Gamma(a): \quad z_1^{a_1}+\cdots
z_n^{a_n}=0. $$ - Brieskorn has proved the following conjecture of Milnor:
$$\Gamma(a)\quad \text{is a topological}
\iff \prod_{1\leq k_l\leq
a_k-1}(1-\epsilon_1^{k_1}
\epsilon_1^{k_2}\cdots
\epsilon_n^{k_n} )=1,$$ where
$\epsilon_k=\mathrm{exp}\Big({\frac{2\pi
}{a_k}\mathrm{i} }\Big)$ for $k=1,\cdots, n$.
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edited Oct 8 2010 at 16:22 |
The simplest example of a singular
algebraic variety which is a
topological manifold is given by the
cusp
$$z_1^2-z_0^3=0.$$
The cusp is
a topological manifold homeomorphic
to a real plane $\mathbb{R}^2$ as can
be seen by the parametrization $t\mapsto (z_1, z_0)= (t^2,t^3)$ where $t$ is a
complex parameter. Munford has proven that a two
dimensional normal complex space
which is a topological manifold is
always nonsingular. Munford's result does not generalize
to (odd) dimensions higher than 2 as
proven by Brieskorn using the
following counter examples which generalizes the case of the cusp:
$$z_1^2+ z_2^2+\cdots
z_{2k+1}^2-z_0^3=0,\quad \text{where} \quad k\in
\mathbb{N}_0.$$ More generally, given $a=(a_1,
\cdots, a_n)\in \mathbb{N}^n_0$ with $a_j>1$ for all $j$, one
can define the following variety
$\Gamma(a)$ known as a
Brieskorn-Pham variety: $$ \Gamma(a): \quad z_1^{a_1}+\cdots
z_n^{a_n}=0. $$ - Brieskorn has proved the following conjecture of Milnor:
$$\Gamma(a)\quad \text{is a topological}
\iff \prod_{1\leq k_l\leq
a_k-1}(1-\epsilon_1^{k_1}
\epsilon_1^{k_2}\cdots
\epsilon_n^{k_n} )=1,$$ where
$\epsilon_k=\mathrm{exp}\Big({\frac{2\pi
}{a_k}\mathrm{i} }\Big)$ for $k=1,\cdots, n$.
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answered Oct 8 2010 at 12:24 |
The simplest example of a singular
algebraic variety which is a
topological manifold is given by the
cusp
$$z_1^2-z_0^3=0.$$
The cusp is
a topological manifold homeomorphic
to a real plane $\mathbb{R}^2$ as can
be seen by the parametrization $t\mapsto (z_1, z_0)= (t^2,t^3)$ where $t$ is a
complex parameter. Munford has proven that a two
dimensional normal complex space
which is a topological manifold is
always nonsingular. Munford's result does not generalize
to (odd) dimensions higher than 2 as
proven by Brieskorn using the
following counter examples which generalizes the case of the cusp:
$$z_1^2+ z_2^2+\cdots
z_{2k+1}^2-z_0^3=0,\quad \text{where} \quad k\in
\mathbb{N}_0.$$
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