A Groebner basis for the 2×2 determinantal ideal mod t2

In an earlier paper [T. Košir, B.A. Sethuraman, Determinantal varieties over truncated polynomial rings, J. Pure Appl. Algebra 195 (2005) 75-95] 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 Zr,km,n obtained by considering generic m×n (m⩽n) matrices over rings of the form F[t]/(tk), 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 I2,2m,n 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 Zr,4m,n where r is arbitrary. (The components of Zr,2m,n and Zr,3m,n were already described in the above cited paper.)