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Let (R,\frak {m}) be a local ring and I, J two arbitrary ideals of R. Let gr_J(R/I) denote the associated graded ring of R/I with respect to J, which corresponds to the normal cone in algebraic geometry. With regards to the finite determinacy of singularity with respect to the Jacobian ideal we study the problem for which ideal I = (f_1,\dots,f_r) does there exist a number N such that if f_i' \equiv f_i \mod J^N, i = 1,\dots,r, and I' = (f_1',\dots,f_r'), then gr_J(R/I) \cong gr_J(R/I'). This problem arises from a recent result of Ma, Quy and Smirnov in the case J is an \frak {m}-primary ideal, which solves a long standing conjecture of Srinivas and Trivedi on the invariance of Hilbert functions under small perturbations. Their approach involves Hilbert functions and cannot be used to study the above general problem. Our main result shows that such a number N exists if f_1,\dots,f_r is locally a regular sequence outside the locus of J. It has interesting applications to a range of related problems. [ABSTRACT FROM AUTHOR]

Introduction Determinantal ideals are one of important topics in Algebraic Geometry and Commutative Algebra. There are several examples of varieties as rational normal scrolls which their defining ideals are generated by minors of a matrix. In general, Grobner bases for an ideals helps us to correspond a monomial ideal to the main ideal with the same invariants in some senses. In this paper, we compute a Grobner bases for an ideal generated by 2-minors of a 2×n matrix of monomials. Main results Let S = k[x1, ..., xn] be the polynomial ring over a field k. Let for 1 ≤ i ≤ n, Xi = {xil, 1 ≤ l ≤ s} for some positive integer s. Let M= [ m11(X1 ) ⋯ m1n (Xn) ⋮ ... ⋮ mt1 (X1) ⋯ mtn(Xn)] be a matrix such that mij(Xj) is a monomial of indeterminates in Xj. A t-minor [r1 r2 ... rt | c1 c2 ... ct] of M is determinant of the submatrix with rows r1 r2 ... rt and columns c1 c2 ... ct. Let It (M) be the ideal generated by all t-minors of M. Let ≤ be a monomial order and the following conditions are satisfied: 1. Each column is decreasing from top to bottom. 2. Each monomial in a column is greater that each monomial in a column in the right. 3. Each two monomials are different. With these assumptions, we have the following theorem. Theorem. For t = 2, set of all 2-minors of M is a Grobner bases for I2 (M). Proposition. Height of It (M) is equal to n-t+1. Conclusion A 2 by n matrix of monomials appears in some topics of algebraic combinatorics and if we omit each one of the above conditions, the theorem might be false. [ABSTRACT FROM AUTHOR]