  In arXiv:math/0405373 , Eisenbud, Huneke and Ulrich conjectured a result on
the Castelnuovo-Mumford regularity of the embedding of a projective space
$\mathbb{P}^{n-1}\hookrightarrow \mathbb{P}^{r-1}$ determined by generators of
a linearly presented $\mathfrak{m}$-primary ideal. This result implies in
particular that the image is scheme defined by equations of degree at most $n$.
In this text we prove that the ideal of maximal minors of the Jacobian dual
matrix associated to the input ideal defines the image as a scheme; it is
generated in degree $n$. Showing that this ideal has a linear resolution would
imply that the conjecture in arXiv:math/0405373 holds. Furthermore, if this
ideal of minors coincides with the one of the image in degree $n$ - what we
hope to be true - the linearity of the resolution of this ideal of maximal
minors is equivalent to the conjecture in arXiv:math/0405373.
