I need a book which contains the demonstration that if $\Gamma$ is a curve in $\mathbb{P}^{n}(\mathbb{R})$ defined as $F(x_{1}, \cdots \ \ , x_{n+1}) = 0 $ , with $F$ homogeneous polynomial, then $\Gamma \cap \mathbb{U}_{i}$ is a differentiable manifold, with $\mathbb{U}_{i} = \{[x_{1}: \ \cdots \ :x_{n+1}] | \ \ x_{i} \neq 0\}$ .
Where can I find it ?
Unless $n=2$ you are not going to get a curve, but rather a manifold of dimension $n-1$. Intuitively, you start with a space of dimension $n$ and you impose one constraint, so you end up with something of dimension $n-1$.
Second, unless the polynomial is non-singular, i.e. its gradient is nowhere zero, you are not going to get a smooth manifold but something which can have singularities like e.g. self-intersection.
Having said that, on each $\mathbb{U_i}$, $x_i\neq 0$, so you can consider the set $\Gamma\cap\mathbb{U}_i=\{(y_1,\ldots \hat{y}_i, \ldots, y_{n})\in\mathbb{R}^{n} : F(y_1,\ldots, 1,\ldots y_{n+1})=0 \}$, where $y_1=x_1/x_i,\ldots$. By the preimage theorem http://en.wikipedia.org/wiki/Preimage_theorem you then have that $\Gamma\cap\mathbb{U}_i$ is a smooth manifold of dimension $n-1$.