Our object of study is a rational map defined by homogeneous forms
$g_{1},\ldots ,g_{n}$
, of the same degree
$d$
, in the homogeneous coordinate ring
$R=k[x_{1},\ldots ,x_{s}]$
of
$\mathbb{P}_{k}^{s-1}$
. Our goal is to relate properties of
$\unicode[STIX]{x1D6F9}$
, of the homogeneous coordinate ring
$A=k[g_{1},\ldots ,g_{n}]$
of the variety parameterized by
$\unicode[STIX]{x1D6F9}$
, and of the Rees algebra
${\mathcal{R}}(I)$
, the bihomogeneous coordinate ring of the graph of
$\unicode[STIX]{x1D6F9}$
. For a regular map
$\unicode[STIX]{x1D6F9}$
, for instance, we prove that
${\mathcal{R}}(I)$
satisfies Serre’s condition
$R_{i}$
, for some
$i>0$
, if and only if
$A$
satisfies
$R_{i-1}$
and
$\unicode[STIX]{x1D6F9}$
is birational onto its image. Thus, in particular,
$\unicode[STIX]{x1D6F9}$
is birational onto its image if and only if
${\mathcal{R}}(I)$
satisfies
$R_{1}$
. Either condition has implications for the shape of the core, namely,
$\text{core}(I)$
is the multiplier ideal of
$I^{s}$
and
$\text{core}(I)=(x_{1},\ldots ,x_{s})^{sd-s+1}.$
Conversely, for
$s=2$
, either equality for the core implies birationality. In addition, by means of the generalized rows of the syzygy matrix of
$g_{1},\ldots ,g_{n}$
, we give an explicit method to reduce the nonbirational case to the birational one when
$s=2$
.