Determinantal varieties over truncated polynomial rings
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 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 the ideal \(I_{r,k}\) 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 (see e.g. \textit{W. Bruns} and \textit{U. Vetter} [``Determinantal rings'', Lect. Notes Math. 1327 (1988; Zbl 0673.13006)]). Let \(m\leq n\). The authors show that \(Z_{r,k}\) is irreducible if and only if \(r= m\). If \(r< m\), then \(Z_{r,k}\) has \(\leq\lfloor k/2\rfloor+ 1\) components with equality for \(r= 2\). In some special cases they determine the components explicitly.
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