On Rees algebras of linearly presented ideals and modules

Let I be a perfect ideal of height two in \(R=k[x_1, \ldots , x_d]\) $R=k[x_1,\dots ,x_d]$ and let \(\varphi \) $\phi$ denote its Hilbert–Burch matrix. When \(\varphi \) $\phi$ has linear entries, the algebraic structure of the Rees algebra \({\mathcal {R}}(I)\) $\mathcal{R}(I)$ is well-understood under the additional assumption that the minimal number of generators of I is bounded locally up to codimension \(d-1\) $d-1$ . In the first part of this article, we determine the defining ideal of \({\mathcal {R}}(I)\) $\mathcal{R}(I)$ under the weaker assumption that such condition holds only up to codimension \(d-2\) $d-2$ , 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.