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.