We work over complex numbers. Let $X \subset \mathbb{P}^n$ a complex projective subvariety associated to homogeneous ideal $I(X) \subset \mathbb{C}[X_0,...,X_n]$. Assume $X$ contains point $p:= (1,0,...,0) \in \mathbb{P}^n$. Denote the homogeneous coordinate ring $\mathbb{C}[X_0,...,X_n]/I(X)$ as $R_X$.
Let $F_0(X), F_1(X),..., F_d(X) \in R_X$ homogeneous elements of degree $k$ of $R_X$ NOT vanishing simultaneously on $X-\{p\}$. Then these $F_j$ define a regular map
$$ f: X-\{p\} \to \mathbb{P}^d, y \mapsto [F_0(x):...: F_d(x)] $$
Question: How can I write down EXPLICITELY an extension of $f$ to $\bar{f}: B_p(X) \to \mathbb{P}^d$, where $B_p(X) $ is the blowup of $X$ at $p=(1,0,...,0)$?
An IDEA:(don't know if that's the most 'natural' way to do it or even if it's well defined at all) By definition $B_p(X)$ equals is contained in closed subset
$$ S :=\{(x,y)= ([x_0: x_1: ... x_n], [y_1:... y_n]) \ \vert \ x \in X, y \in x_i y_j - x_j y_i \text{ for } $1 \le i,j \le n\} \subset \mathbb{P}^n \times \mathbb{P}^{n-1} $$
Therefore we have an inclusion of ideals $I(S) \subset I(B_p(X))
\subset \mathbb{C}[X_0, ..., X_n, Y_1,..., Y_n]$ where the $X_i$ are
the coordinate functions of $\mathbb{P}^n $ and $Y_j$ of $\mathbb{P}^{n-1}$.
If we establish an extension of $f$ to a map
$g: S \to \mathbb{P}^d$, then this gives us also an extension
to $B_p(X)$ since $S$ contains it as closed subvariety.
Now is it possible to extend $f$ to $S$ by doing following: Every $F_j(X) \in R_X$ is represented by a (not unique) homogeneous element $G_j(X) \in \mathbb{C}[X_0,...,X_n]$ of degree $k$. Therefore we can write $G_j(X)= \sum_{\vert I \vert =k} g_{j}^I X^I $ ($I$ multiindex in $(i_0,... i_n)$)
Then $f$ can be also writen as $y \mapsto [G_0(x):...: G_d(x)] $.
Now my idea to extend $f$ to $h$ was to replace in every
$G_j(X)= \sum_{\vert I \vert =k} g_{j}^I X^I $ every $X_i $ with $0 <i \le n$ by
$Y_i$ from coordinate ring of $\mathbb{P}^{n-1}$. We obtain
$$H_j(X,Y) := \sum_{I= (i_0, \widetilde{I}): \vert I \vert =k} g_{j}^I X_0^{i_0}Y^{\widetilde{I}} $$
and we set
$$ h: S \to \mathbb{P}^d, y \mapsto [H_0(y):...: H_d(y)] $$
which by construction should restrict to $X-\{p\}$, since for $x=[x_0: x_1: ... x_n] \in X-\{p\}$ there exists a unique $y =[y_1:... y_n]$ with $(x,y) \in S$ due to relation above.
Therefore $h$ seems to be welldefined and extends naturally $X-\{p\}$. Therefore it's
restriction to $B_p(X) \subset S$ should be the desired extension.
Does my construction work or is there are more reasonable "natural textbook method" known to reach
the extension of $f$ to $B_p(X)$?
(A note: I know that there is a formal result known that after sufficient times of blowing up a rational map be obtain a regular map, but here I want to discuss the explicit construction of such lift)