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

Abstract

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 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 m,n r,4 where r is arbitrary. (The components of Z m,n r,2 and Z m,n r,3 were already described in the above cited paper.) 

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“…We have noted in the introduction that a Gröbner basis (with respect to reverse lexicographic order on monomials with the 2mn variables arranged suitably) of the ideal Ᏽ of the variety ᐆ m,n 2,2 of first-order jets over ᐆ m,n 2 , as well as of the ideal Ᏽ 0 of the principal component Z 0 of ᐆ m,n 2,2 , was determined in [Košir and Sethuraman 2005b]. As a consequence, one can write down the generators of the leading term ideal of Ᏽ 0 (see [Jonov 2011, Proposition 1.1]), say LT(Ᏽ 0 ), and deduce that R/LT(Ᏽ 0 ) is the Stanley-Reisner ring of a simplicial complex 0 with V as its set of vertices.…”

“…Proof. It is well-known that the (Krull) dimension as well as the Hilbert series of R/Ᏽ 0 coincides with that of R/LT(Ᏽ 0 ) (see, e.g., [Bruns and Conca 2003, §3]), where LT(Ᏽ 0 ) denotes the leading term ideal of Ᏽ 0 as in [Košir and Sethuraman 2005b] and [Jonov 2011, Proposition 1.1]. Now 0 is precisely the simplicial complex such that R/LT(Ᏽ 0 ) is the Stanley-Reisner ring of 0 .…”

“…Now determinantal varieties such as ᐆ m,n r above are natural examples of singular algebraic varieties, and it is not surprising that the study of their jet schemes has been of considerable interest. This was done first by Košir and Sethuraman [2005a;2005b] (see also [Yuen 2007a]). To describe the related results, henceforth we fix positive integers r, k, m, n with r ≤ m ≤ n, and let ᐆ m,n r,k denote the (k − 1)-th jet scheme on ᐆ m,n r .…”

“…2,2 and Ᏽ 0 denote, respectively, the ideals of R corresponding to the jet scheme ᐆ m,n 2,2 and its principal component Z 0 . In [Košir and Sethuraman 2005b], it was shown that both Ᏽ and Ᏽ 0 are homogeneous radical ideals of R (so that Ᏽ 0 is prime), and moreover their Gröbner bases were explicitly determined. The leading term ideal LT(Ᏽ 0 ) of Ᏽ 0 with respect to this Gröbner basis is generated by squarefree monomials and hence R/LT(Ᏽ 0 ) is the Stanley-Reisner ring of a simplicial complex 0 .…”