Prime saturations and rees algebras of almost linearly presented ideals

Let I be a height two perfect ideal in the polynomial ring k[x1,…,x d] satisfying the Gd condition. Suppose I admits a homogeneous presentation matrix composed of linear columns except for one column of degree n. In this setting, we give two descriptions of the ideal A defining the Rees algebra Rees(I) and if in addition I is generated by d+1 elements, we give an explicit generating set for A. In aid of these results, we study saturations in a more general setting. We show that, with some assumptions, a saturation being a prime ideal attaining its maximum possible height is a property that can be passed to a smaller saturation. Using this, we prove that in a polynomial ring over a Cohen-Macaulay ring containing a field, a saturation being a prime ideal attaining its maximum possible height often implies that it is a geometric residual intersection