On the Cohen–Macaulayness of tangent cones of monomial curves in $A^4(K)$
On the Cohen–Macaulayness of tangent cones of monomial curves in $A^4(K)$
In this paper we give necessary and sufficient conditions for the Cohen–Macaulayness of the tangent cone ofa monomial curve in 4-dimensional affine space. We particularly study the case where C is a Gorenstein noncompleteintersection monomial curve and we generalize some results in the literature. Moreover, by using these results, weconstruct families supporting Rossi’s conjecture, which is still open for monomial curves in 4-dimensional affine space.
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- [1] Arslan F, Mete P. Hilbert functions of Gorenstein monomial curves. Proceedings of the American Mathematical
Society 2007; 135: 1993-2002.
- [2] Barucci V, Fröberg R. Associated graded rings of one-dimensional analytically irreducible rings. Journal of Algebra
2006; 304: 349-358.
- [3] Bayer D, Stillman M. Computation of Hilbert functions. Journal of Symbolic Computation 1992; 14: 31-50.
- [4] Bresinsky H. Symmetric semigroups of integers generated by 4 elements. Manuscripta Mathematica 1975; 17 (3):
205-219.
- [5] Garcia A. Cohen-Macaulayness of the associated graded of a semigroup rings. Communications in Algebra 1982;
10: 393-415.
- [6] Gimenez P, Srinivasan H. A note on Gorenstein monomial curves. Bulletin of the Brazilian Mathematical Society,
New Series 2014; 4: 671-678.
- [7] Herzog J. Generators and relations of abelian semigroups and semigroup rings. Manuscripta Mathematica 1970; 3:
175-193.
- [8] Herzog J. When is a regular sequence super regular? Nagoya Mathematical Journal 1981; 83: 183-195.
- [9] Herzog J, Stamate DI. On the defining equations of the tangent cone of a numerical semigroup ring. Journal of
Algebra 2014; 418: 8-28.
- [10] Jafari R, Zarzuela S. On monomial curves obtained by gluing. Semigroup Forum 2014; 2: 397-416.
- [11] Katsabekis A, Ojeda I. An indispensable classification of monomial curves in A4(K) . Pacific Journal of Mathematics
2014; 268: 95-116.
- [12] Kunz E. The value semigroup of one dimensional Gorenstein ring. Proceedings of the American Mathematical
Society 1970; 25: 748-751.
- [13] Molinelli S, Tamone G. On the Hilbert function of certain rings of monomial curves. Journal of Pure and Applied
Algebra 1995; 101: 191-206.
- [14] Oneto A, Strazzanti F, Tamone G. One-dimensional Gorenstein local rings with decreasing Hilbert function. Journal
of Algebra 2017; 489: 91-114.
- [15] Patil DP, Tamone G. CM defect and Hilbert functions of monomial curves. Journal of Pure and Applied Algebra
2011; 215 (7): 1539-1551.
- [16] Pisón Casares P. The short resolution of a lattice ideal. Proceedings of the American Mathematical Society 2003;
131: 1081-1091.
- [17] Rossi M. Hilbert functions of Cohen-Macaulay local rings. Commutative Algebra and its Connections to Geometry,
Contemporary Mathematics 2011; 555: 173-200.
- [18] Şahin M, Şahin N. On pseudo symmetric monomial curves. Communications in Algebra 2018; 46 (6): 2561-2573.
- [19] Sharifan L, Zaare-Nahandi R. Minimal free resolution of the associated graded ring of monomial curves of generalized
arithmetic sequences. Journal of Pure and Applied Algebra 2009; 213: 360-369.
- [20] Sturmfels B. Gröbner Bases and Convex Polytopes. University Lecture Series Vol. 8. Providence, RI, USA: American
Mathematical Society, 1996.