5 Improved formatting (deleted the bug enter your code here)

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• 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 $k=1,\cdots, n$.
4 Typos: Mumford; manifold.

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 $k=1,\cdots, n$.
3 Typos

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 $k=1,\cdots, n$.