Tight closure of powers of parameter ideals in hypersurface rings and\n their tight Hilbert polynomials

2021/06/16 by Saipriya Dubey, Dubey, Saipriya, Vivek Mukundan +3
Mathematics · Computer Science · #Commutative Algebra and Its Applications #Algebraic Geometry and Number Theory #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2106.08825

Abstract

In this paper we find the tight closure of powers of parameter ideals of\ncertain diagonal hypersurface rings. In many cases the associated graded ring\nwith respect to tight closure filtration turns out to be Cohen-Macaulay. This\nhelps us find the tight Hilbert polynomial in these diagonal hypersurfaces. We\ndetermine the tight Hilbert polynomial in the following cases: (1) F-pure\ndiagonal hypersurfaces where number of variables is equal to the degree of\ndefining equation, (2) diagonal hypersurface rings where characteristic of the\nring is one less than the degree of defining equation and (3) quartic diagonal\nhypersurface in four variables.\n

Related

• A computation of tight closure in diagonal hypersurfaces
• An inequality involving tight closure and parameter ideals
• Associated graded rings of the filtration of tight closure of powers of parameter ideals
• Depth of associated graded rings via Hilbert coefficients of ideals
• How to compute the Hilbert depth of a graded ideal