Standard Cremona Involution Let $\varphi$ be the standard Cremona involution on $\mathbb{P}^r$, which is defined as $[x_0,\dots,x_r]\mapsto [\frac{x_0\dots x_r}{x_0},\dots, \frac{x_0\dots x_r}{x_r} ]$. 
I came across to the fact that a rational normal curve passing through all coordinate points (i.e $[0,\dots,0,1,0,\dots,0]$) maps to a line under $\varphi$. I could confirm this by writing down equations for the curve and applying $\varphi$, but I feel that there should be a cleaner way to do it, which should give a similar statement for any variety $X\subseteq \mathbb{P}^r$ (not just rational normal curves). So, my first question is:  
i) Given a variety $X\subseteq\mathbb{P}^r$ of some degree $d$, what can we say about the degree of $\varphi(X)$?
I have seen in some papers (without much explanation) that $\varphi$ is the same as the map given by the linear system of degree $r$ hypersurfaces which vanish with multiplicity $r-1$ at coordinate points. 
ii) How can I see that this alternative description gives the same map $\varphi$?
Any help will be appreciated.
 A: To compute the degree of $\varphi(X)$ we need to cut it by generic hyperplanes $H_1,\cdots,H_k$, $k=\dim(X)$ and count the number of points in such generic intersection. Then notice that this is the same as intersecting $X$ with the pullback of the hyperplanes $\varphi^*H_i$, $i=1,\cdots,k$. So the computation of $\deg(\varphi(X))$ is reconduced to an appropiated counting of intersections between $X$ and $k$ degree $r$ hypersurfaces which vanish with multiplicity $r−1$ at coordinate points. Then it is clear that you need to start considering two possibilites: either some of the $r+1$ coordinates points belong to $X$ or no one of them belongs to $X$. In the last case by Bezout theorem $\deg(\varphi(X)) = \deg(X).r^k =\deg(X)r^{\dim(X)}.$
If some of the $r+1$ coordinates points belong to $X$ then it is clear that $\deg(\varphi(X))$ is going to be less than $\deg(X)r^{\dim(X)}$ because this points are not really in the image $\varphi(X)$. So we need to count with multiplicities the intersection of $X$ and the hypersurfaces $\varphi^*H_i$ to know how many of such points we have to delete at $\deg(X)r^{\dim(X)}$.  For example, for the canonical normal curve $C_r$ passing through all $r+1$ points we check that the multiplicies are $r-1$ at each of them. Hence $$ \deg(\varphi(C_r)) = \deg(X)r^{\dim(X)}- (r+1)(r-1) = r.r - (r+1)(r-1) = 1 \, .$$ Namely, the image $\varphi(C_r)$ is a line.
For your second question just notice that a mapa $f$ from $\mathbb{P}^r$ to $\mathbb{P}^r$ are defined by $(r+1)$ homogeneous degree $d$ polynomials of $f_0,f_1,f_2,\cdots,f_r$ as $$f(x) = [f_0(x):f_1(x):\cdots:f_r(x)]$$
usually the map $f$ is considered up to change of coordinates of the target  $\mathbb{P}^r$. Namely, you agree in consider $f(x)$ to be the same map as $\tilde{f}(x)$ where $\tilde{f}(x) = [\sum_{i=0}^r a_{i0}f_i : \cdots : \sum_{i=0}^r a_{ir}f_i]$    where $A = (a_{ij})$ is a $(r+1)\times(r+1)$ invertible matrix i.e. a change of coordinates in the target $\mathbb{P}^r$.
Then notice that doing so you are considering all possible linear combinations $$\sum_{i=0}^r \lambda_ i f_i$$ of your $f_i$ to be used as the components of the map $f:\mathbb{P}^r \to \mathbb{P}^r$. Such linear combinations of the $f_i$ is, by definition, what you called linear system. Usually, the interesting linear systems are defined by an interesting linear property e.g. all degree $r$ hypersurfaces which vanish with multiplicity (r−1) at coordinate points.
There is a lot of literature in Internet about Cremona transformations, so I am not going to give any reference here. 
