A Gröbner basis for the \(2\times 2\) determinantal ideal mod \(t^2\) (Q2576199)

Consider the polynomial ring \(F[t]\) in a variable \(t\) over a field \(F\), and for some positive integer \(k\) the ring \(R= F[t]/(t^k)\). \(X(t)\) is defined as the ``generic'' \(m\times n\)-matrix \((m\leq n)\) over \(R\) which means that the \((i, j)\)-entry of \(X(t)\) is a truncated polynomial \(x^{(0)}_{ij}+ x^{(1)}_{ij}t+\cdots+ x^{(k-1)}_{ij} t^{k-1}\) where the \(x^l_{ij}\), \(1\leq i\leq m\), \(1\leq j\leq n\), \(0\leq l< k\) are distinct indeterminates. (So \(X(t)\) is an \(m\times n\)-matrix over the ring \(F[x^l_{ij}\mid 1\leq i\leq m\), \(1\leq j\leq n\), \(0\leq l< k][t]/(t^k)\).) For a fixed \(r\) consider the \(r\)-minors of \(X(t)\) and let \(I_{r,k}\) be the ideal in \(F[x^l_{ij}]\) generated by the coefficients of all these \(r\)-minors. In case \(F\) is algebraically closed, \(Z_{r,k}\) denotes the variety in \(F^{mnk}\) defined by \(I_{r,k}\). For \(k= 1\) one obtains the classical determinantal ideals and varieties. In a previous paper [J. Pure Appl. Algebra 195, 75--95 (2005; Zbl 1085.14043)] the authors studied the components of \(Z_{r,k}\). They determined them in the cases in which \(k= 2\) or \(k= 3\). In the paper under discussion they compute a Gröbner basis of the ideal \(I_{2,2}\) and use their results to describe the components of \(Z_{r,4}\).