Let $r: \mathbb{P}^n_{\mathbb{C}} \dashrightarrow \mathbb{P}^n_{\mathbb{C}} $ be a birational map
$$ [X_0: X_1: \cdots : X_n] \mapsto [F_0(\overline{X}): \cdots F_n(\overline{X})]$$
where $F_i(\overline{X})$ are homogeneous in $X_0,..., X_n$ of degree $d$. Let $BL_r$ the base locus where $r$ is not defined; that is the vanishing set of all $F_i$. Because $r$ is birational it is well defined on a nonempty Zariski open set $\mathbb{P}^n_{\mathbb{C}} - BL_r = \bigcup_i D_+(F_i)$.
Let $D= V_+(H(\overline{X}))$ be a divisor defined as the vanishing set of a homogeneous polynomial $H$ of degree $h$.
In Janos Kollar's Resolution of Singularities on pages 36/37 is used (without giving concrete definition) the notation of $r_*D$; the direct image of a divisor wrt $r$.
If we start with explicite description
$ D= V_+(H(\overline{X}))$, does the direct image
$r_*D$ also have an explicite description
in vanishing polynomials?
Note, that the pullback of the divisor $r^*D$ is the vanishing set
of the polynomial $H(F_0,F_1,..., F_n)$, a homogeneous polynomial of degree $hd$. Is there a similar well known formula for $r_*D$?
$r_*D$ is the birational transform of $D$, which means, it is the closure of $r(D-BL_r)$ in the target $\mathbb{P}^n_{\mathbb{C}}$.