A note on d -sequences and regular sequences of quadrics

Let K be a field and X, Y denote matrices such that the entries of X are either indeterminates over K or 0, not all zero, and the entries of Y are indeterminates over K which are different from those appearing in X. We consider the ideal I<inf>1</inf>(XY) , which is the ideal generated by the homogeneous polynomials of degree 2 given by the 1 × 1 minors of the matrix XY. We prove that d-sequences and regular sequences arise naturally as part of generators of I<inf>1</inf>(XY) for some special cases. We use this information to calculate the equations defining the Rees algebra of I<inf>1</inf>(XY).