Box-shaped matrices and the defining ideal of certain blowup surfaces

In this paper, we generalize the notions of a matrix and its ideal of 2×2 minors to that of a box-shaped matrix and its ideal of 2×2 minors, and make use of these notions to study projective embeddings of certain blowup surfaces. We prove that the ideal of 2×2 minors of a generic box-shaped matrix is a perfect prime ideal that gives the algebraic description for the Segre embedding of the product of several projective spaces. We use the notion of the ideal of 2×2 minors of a box-shaped matrix to give an explicit description for the defining ideal of the blowup of image along a set of image points in generic position, embedded into projective spaces using very ample divisors which correspond to the linear systems of plane curves going through these points.