Monomial Curves in Afinne Space and their Associated Prime Ideals with Six Generators as Set-Theoretic Complete Intersections

Michaela Holesova

doi:10.26552/com.c.2014.3.65-68

Abstract

The paper deals with the problem of the expression of associated prime ideals of monomial curves in the affine space A4 as set-theoretic complete intersections. We describe some associated prime ideals, a minimal generating set of which has six elements and we prove that these ideals are set-theoretic complete intersections. Corresponding monomial curves are intersections of three hypersurfaces and we find the equations of these hypersurfaces.

Highlights

It is known that k-dimensional algebraic affine variety is intersection of not fewer than n - k hypersurfaces in n-dimensional affine space An
There is the presumption, that a number of these hypersurfaces is exactly n - k. In this case we can say, that they are ideal-theoretic or set-theoretic complete intersections. This is equivalent to the fact, that either the associated ideal I of this variety has generators or the ideal I is radical of an ideal a, a 3 I, the ideal a has n - k generators
We showed that associated prime ideals of monomial curves whose minimal set of generators have five elements is s.t.c.i. [7]

Summary

Introduction

There is the presumption, that a number of these hypersurfaces is exactly n - k In this case we can say, that they are ideal-theoretic or set-theoretic complete intersections. R such that f^t ,n1 tn, tn3, tn4h = 0, t transcendental over K, is the associated prime ideal of ring R of the monomial curve C. Bresinsky proved that if numerical semigroup H is symmetric, the monomial curve C^n1, n2, n3, n4h and its associated prime ideals are s.t.c.i.(see [2]). W. Gastinger in [6] proved that associated prime ideals of monomial curves in A4 are s.t.c.i. if minimal generating sets of these ideals have four generators. We showed that associated prime ideals of monomial curves whose minimal set of generators have five elements is s.t.c.i. We showed that associated prime ideals of monomial curves whose minimal set of generators have five elements is s.t.c.i. [7]

The associated prime ideal P of the monomial curve C

The minimal generating set for a prime ideal P having six generators

Set-theoretic complete intersection