  Our object of study is a rational map Psi from projective s-1 space to
projective n-1 space defined by homogeneous forms g1,...,gn, of the same degree
d, in the homogeneous coordinate ring R=k[x1,...,xs] of projective s-1 space.
Our goal is to relate properties of Psi, of the homogeneous coordinate ring
A=k[g1,...,gn] of the variety parametrized by Psi, and of the Rees algebra
R[It], the bihomogeneous coordinate ring of the graph of Psi. For a regular map
Psi, for instance, we prove that R[It] satisfies Serre's condition R_i, for
some positive i, if and only if A satisfies R_{i-1} and Psi is birational onto
its image. Thus, in particular, Psi is birational onto its image if and only if
R[It] satisfies R_1. Either condition has implications for the shape of the
core, namely, the core of I is the multiplier ideal of I to the power s and the
core of I equals the maximal homogeneous ideal of R to the power sd-s+1.
Conversely, for s equal to two, either equality for the core implies
birationality. In addition, by means of the generalized rows of the syzygy
matrix of g1,...,gn, we give an explicit method to reduce the non-birational
case to the birational one when s is equal to 2.
