Let $f:\mathbb{P}^n\_1\dashrightarrow\mathbb{P}^n\_2$ be the standard Cremona transformation based on $p\_1,...,p\_{n+1}\in\mathbb{P}^n\_1$ and $q\_1,...,q\_{n+1}\in\mathbb{P}^n\_2$. That is, $f$ is the rational map associated to the linear system of hypersurfaces of degree $n$ having multiplicity $n-1$ at $p\_1,...,p\_{n+1}$.


Let $D\subset\mathbb{P}^n$ be an hypersurface of degree $d$ having multiplicities $m\_1,...,m\_{n+1}$ at $p\_1,...,p\_{n+1}$. Let us assume that $D$ is not contracted by $f$. Is there a formula for the degree of $f(D)$, and for the multiplicities of $f(D)$ at $q\_1,...,q\_{n+1}$?


For instance, by an explicit computation, I got that when $d = 1$ and $D$ is an hyperplane spanned by $n-2$ of the $p\_i$'s and another two general points then $f(D)$ is an irreducible quadric whit a codimension three linear space as vertex.



Cremona transformations
For simplicity of notation we may assume $\mathbb{P}^n\_1=\mathbb{P}^n\_2=\mathbb{P}^n$, and $q\_i=p\_i$. Let $P$ be the variety obtained by blowing up $\mathbb{P}^n$ at $p\_1,\ldots ,p\_{n+1}$, $E\_i$ the exceptional divisor above $p\_i$, $H$ the pull back of a hyperplane section. Then $f$ induces an involution of $P$ (which I still denote $f$), and one has in $\mathrm{Pic}(P)$ :
$$f\_\*h=nH-\sum (n-1)E\_i\qquad f\_\*E\_i=H-\sum\_{j\neq i}E\_j$$
By computing 
$f\_\*(dH-\sum m\_iE\_i)$ you get $\deg f(D)=dn-\sum m\_i$ and $\mathrm{mult}\_{p\_i}f(D)=d(n-1)-\sum\_{j\neq i}m\_j$.


Note that your final computation is incorrect, the hyperplane through $n$ of the $p\_i$ is contracted to a point (think of the standard Cremona transformation for $n=2$).


