Abstract. Consider a grade 2 perfect idealwhich is generated by forms of the same degree. Assume that the presentation matrix ϕ is almost linear, that is, all but the last column of ϕ consist of entries which are linear. For such ideals, we find explicit forms of the defining ideal of the Rees algebra R(I). We also introduce the notion of iterated Jacobian duals.

We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. One criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning when the third symbolic power is in the square of an ideal, and a strong resurgence bound for some classes of space monomial curves.

Let (R, m) be an analytically unramified local ring of positive prime characteristic p.For an ideal I, let I * denote its tight closure. We introduce the tight Hilbert function H * I (n) = ℓ(R/(I n ) * ) and the corresponding tight Hilbert polynomial P * I (n), where I is an m-primary ideal. It is proved that F -rationality can be detected by the vanishing of the first coefficient of P * I (n). We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes. IntroductionLet (R, m) be a d-dimensional Noetherian local ring and I be an m-primary ideal. Let I be the integral closure of I. The Rees algebra of I is denoted byWe use ℓ(M ) to denote the length of an R-module M. David Rees [14] showed that if R is analytically unramified then R(I) is a finite module over R(I). This implies that for all large n, the normal Hilbert function of I, H I (n) = ℓ(R/I n ) is a polynomial of degree d. This is called the normal Hilbert polynomial of I and it is denoted by P I (n). We write P I (n) = e 0 (I)Here e 0 (I) = e(I), the multiplicity of I and the coefficients e i (I) for i = 0, 1, . . . , d are called the normal Hilbert coefficients of I. The normal Hilbert coefficients play an important role in the study of singularities of algebraic varieties. Rees [15] proved that if (R, m) is a 2-dimensional analytically unramified normal local ring then it is pseudo-rational if and only if e 2 (I) = 0 for all m-primary ideals I of R. Shiroh Itoh [9] proved that if R is Cohen-Macaulay and analytically unramified, then e 3 (I) ≥ 0. Moreover, if R is Gorenstein and I is generated by a regular sequence so that I = m, then e 3 (I) = 0 if and only if r(I) ≤ 2. Here r(I) = min{n | II n = I n+1 }. The integer r(I) is called the normal reduction number of I. Itoh [9] proved that if r(I) ≤ 2, then R(I) is Cohen-Macaulay. He

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In this paper, we study the depth and the Castelnuovo–Mumford regularity of the powers of edge ideals which are unmixed and whose underlying graphs are bipartite. In particular, we prove that the depth of the powers of the edge ideal stabilizes when the exponent is the same as half the number of vertices in the underlying connected bipartite graph. We also define the idea of “drop” in the sequence of depth of powers of ideals. Further, we show that the sequence of depth of the powers of such edge ideals may have any number of “drops”. In the process of proving these results we put forward some interesting examples and some questions for future research. As for regularity, we establish a formula for the regularity of the powers of such edge ideals in terms of the regularity of the edge ideal itself.