Initial complex associated to a jet scheme of a determinantal variety (Q626720)

Consider a classical determinantal variety \(X\) of \(m\times n\) (\(m\leq n\)) matrices of rank at most~1. It is known [\textit{T. Košir} and \textit{B. A. Sethuraman}, J. Pure Appl. Algebra 195, No. 1, 75--95 (2005; Zbl 1085.14043), J. Algebra 292, No. 1, 138--153 (2005; Zbl 1105.13009)] that, for \(m\geq 3\), the first-order jet scheme over \(X\) has two components \(Y\) and \(Y'\), where \(Y'\) is simply the affine space \(\mathbb A^{mn}_F\) and \(Y\) is the closure of the set of tangents at the nonsingular points of \(X\). This paper under review shows that the coordinate ring of \(Y\) is Cohen-Macaulay (Theorem~1.2). The author proves this in the following way. The Gröbner basis of the defining ideal \(J\) of \(Y\) for the graded lexicographical order has been already known [see loc. cit.] and we know that the ideal \(LT(J)\) of its leading terms is a squarefree monomial ideal, i.e., a Stanley-Reisner ideal. Then the author describes the structure of all facets of the simplicial complex \(\Delta_{LT(J)}\), corresponding to \(LT(J)\) (Thereom~2.5). In particular, it is shown that \(\Delta_{LT(J)}\) is shellable (Theorem~3.2). This implies that quotient ring by \(LT(J)\) and thus by \(J\) is Cohen-Macaulay.