Abstract: The search for the defining equations of Rees algebras is a classical problem within commutative algebra and algebraic geometry. Often called the blowup algebra, the Rees ring provides an algebraic insight into the blowup of a variety along a subvariety. In this talk, we resort to the classical setting of the Rees ring of a codimension-two perfect ideal in $R=k[x_1,\ldots,x_d]$, with linear presentation. Traditionally, one imposes the so-called Artin-Nagata property $G_d$ on the number of generators of the ideal in certain codimensions, however we proceed assuming the weaker condition $G_{d-1}$. By employing a modification to the Jacobian dual method, the equations defining the Rees algebra and the special fiber ring are determined, generalizing the work of P.H.L Nguyen. We conclude by posing questions and directions for future work under the condition $G_s$ for $s\leq d-1$. This is part of joint work with Alessandra Costantini and Edward Price.
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