Let $(X_0,X_1,...,X_n)$ homogeneous coordinates of $\mathbb{P}^n$ and let assume that $X^r \subset \mathbb{P}^n$ is a complex variety where $x:= (1,0,...,0) \in X$ and $X$ isn't a cone with vertex $x$.
Consider the graded ring homomorphism
$$ \phi: \mathbb{C}[X_0,...,X_n] \to \oplus_{k=0} O_{x, \mathbb{P}^n} \cdot X_0^k $$
given by $\phi(f):= (f/X_0^k) \cdot X_0^k$ if $f$ is homogeneous of degree $k$. It induced the graded homomorphism
$$ \psi:R_X= \mathbb{C}[X_0,...,X_n]/I(X) \to \oplus_{k=0} O_{x, X} \cdot X_0^k $$
Define as $R^0_X := \psi^{-1}[\oplus_{k=0} m^k_{x, X} \cdot X_0^k]$ where $m_{x, X} $ is the unique maximal ideal of the local ring $\subset O_{x,X}$.
Let $f \in R^0_X$ be homogeneous of degree $l$. Consider the embedding of $X-\{x\}$ into $\mathbb{P}^{N+1}$ defined by monomials in $X_1,..., X_n$ of degree $l$ and $f$. Less formally the map is given by
$$ y \mapsto (X_1^l(x): ... :X_0^{l_0} \cdot X_1^{l_1} ... \cdot X_n^{l_n}(y): ...: f(y)) $$
with $\sum_i l_i =l$. Restricted to $X-\{x\}$ this map is regular.
Now let $B_x(X)$ be the blowup of $X$ in $x$. Geometrically $B_x(X)$ arise also as the Zariski closure of the graph $\Gamma_x^X = \{(y, p_x(y)) \ \vert \ y \in X-\{x\} \} \subset \mathbb{P}^n \times \mathbb{P}^{n-1}$ where the graph $\Gamma_x^X$ is associated to restriction of the projection from $x$ map $p_x: \mathbb{P}^{n} \to \mathbb{P}^{n-1}, (x_0:x_1:...:x_n) \mapsto (x_1:...:x_n)$ to $X-\{x\}$.
Question: why the regular (regular means here well defined in projective sense) embedding map $e: X-\{x\} \to \mathbb{P}^{N+1}$ from above extends to a regular map from $B_x(X)$ to $\mathbb{P}^{N+1}$?
Source: David Mumford's Algebraic Geometry 1: Complex projective varieties. to avoid the clumsiness of having misread something below I attatched to original source:
Here is my attept to solve my problem, is it ok?
The blowup $B_x(X)$ of $X$ in $x $ equals (? or is at least included in the set
$$ \{([x_0: x_1: ... x_n], [y_1:... y_n]) \ \vert \ x \in X, x_i y_j - x_j y_i \} \subset \mathbb{P}^n \times \mathbb{P}^{n-1} $$
We have $X_0, X_1,..., X_n$ as coordinate functions of left factor $\mathbb{P}^n$ and $Y_1,..., Y_n$ as cofu of right factor $\mathbb{P}^{n-1}$.
Now it looks reasonable for me to extend the map $X-\{x\} \to \mathbb{P}^{N+1}$ given by
$$y \mapsto (X_1^l(x): ... :X_0^{l_0} \cdot X_1^{l_1} ... \cdot X_n^{l_n}(y): ...: f(y))$$
by replacing all $X_i$ with $0<i \ge n$ by $Y_i$. This looks like an extension of map from $X-\{x\}$ above because by definition for every $[x_0: x_1: ... x_n] \in X - \{[1:0:...:0]\}$ there is inique $[y_1: ... y_n] \in \mathbb{P}^{n-1}$ with $([x_0: x_1: ... x_n], [y_1:... y_n]) \in B_x(X)$ and it is also welldefined in $\{ x \} \times \mathbb{P}^{n-1}$.
Is the 'solution' acceptable? Unsolved question: is the set $ \{([x_0: x_1: ... x_n], [y_1:... y_n]) \ \vert \ x \in X, x_i y_j - x_j y_i \} \subset \mathbb{P}^n \times \mathbb{P}^{n-1} $ equal to the blowup $B_x(X)$ of $X$ in $x$ or does it only contain $B_x(X)$ properly? More precisely does the fiber of natural map $B_x(X) \to X$ or equal to $\{ x \} \times \mathbb{P}^{n-1}$?