On Rees algebras of linearly presented ideals and modules

Let $I$ be a perfect ideal of height two in $R=k[x_1, \ldots, x_d]$ and let $\varphi$ denote its Hilbert-Burch matrix. When $\varphi$ has linear entries, the algebraic structure of the Rees algebra $\mathcal{R}(I)$ is well-understood under the additional assumption that the minimal number of generators of $I$ is bounded locally up to codimension $d-1$. In the first part of this article, we determine the defining ideal of $\mathcal{R}(I)$ under the weaker assumption that such condition holds only up to codimension $d-2$, generalizing previous work of P.~H.~L.~Nguyen. In the second part, we use generic Bourbaki ideals to extend our findings to Rees algebras of linearly presented modules of projective dimension one.