Determinantal varieties over truncated polynomial rings

Abstract

We study components and dimensions of higher-order determinantal varieties obtained by considering generic m × n (m 6 n) matrices over rings of the form F[t]=(t k ), and for some ÿxed r, setting the coe cients of powers of t of all r × r minors to zero. These varieties can be interpreted as spaces of (k − 1)th order jets over the classical determinantal varieties; a special case of these varieties ÿrst appeared in a problem in commuting matrices. We show that when r = m, the varieties are irreducible, but when r ¡ m, these varieties are reducible. We show that when r = 2 ¡ m (any k), there are exactly k=2 + 1 components, which we determine explicitly, and for general r ¡ m, we show there are at least k=2 + 1 components. We also determine the components explicitly for k = 2 and 3 for all values of r (for k = 3 for all but ÿnitely many pairs of (m; n)).

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

“…A Groebner basis for I = I m,n 2,2 (2 m n) with respect to the grevlex ordering described above consists of the five families of polynomials: consists of all δ [1,2][k,l] and all ε [1,2][k,l] with 1 k < l n, and all ρ [1,2,2][p,q,r] with 1 p < q < r n. (In particular, when m = n = 2, this is a special case of [6,Theorem 3.3] that the defining polynomials δ [1,2] [1,2] and ε [1,2] [1,2] form a Groebner basis for I 2,2 2,2 . This is an exceptional situation, see Remark 2.3 ahead.…”

“…By a variety in A k F we will mean the zero set of a collection of polynomials over F in k variables; in particular, our varieties are not assumed irreducible. In the paper [6], we had begun a study of the components and dimensions of the following varieties that are very closely related to the classical determinantal varieties [2]: Consider the truncated polynomial ring F [t]/(t k ) (k = 1, 2, 3, . .…”

“…We had shown in [6] that Z m,n r,k is irreducible when r = m, and reducible (with at least 1 + k/2 components) when r < m. When r = 2 < m, we had explicitly determined the components, and shown that there are exactly 1 + k/2 of them.…”

“…The two Groebner bases together show that the ideal I 0 is precisely the ideal of one of the two components of Z m,n 2,2 . We then use this result, along with general facts about Z m,n r,k from [6], to provide a complete description of the components of Z m,n r,k in the case k = 4 and r arbitrary. (The components for k = 2 and k = 3 were already described in [6].…”

Which Schubert varieties are local complete intersections?

“…(This is only a resolution of singularities in a weak sense since the image of the exceptional locus contains nonsingular points.) The jet schemes for the special case of determinantal varieties has already been worked out by Mustata [Mus01], Yuen [Yue07], Kosir and Setharuman [KS05], and Docampo [Doc11].…”