Revised Version A GROEBNER BASIS FOR THE 2 × 2

In an earlier paper ([6]) we had begun a study of the components and dimensions of the spaces of (k − 1)-th order jets of the classical determinantal varieties: these are the varieties Z m,n r,k obtained by considering generic m × n (m ≤ n) matrices over rings of the form F [t]/(t k), and for some fixed r, setting the coefficients of powers of t of all r × r minors to zero. In this paper, we consider the case where r = k = 2, and provide a Groebner basis for the ideal I m,n 2,2 which defines the tangent bundle to the classical 2 × 2 determinantal variety. We use the results of these Groebner basis calculations to describe the components of the varieties Z