On the Betti numbers of the tangent cones for Gorenstein monomial curves

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

The aim of the article is to study the Betti numbers of the tangent cone of Gorenstein monomial curves in affine 4-space. If CS is a noncomplete intersection Gorenstein monomial curve whose tangent cone is Cohen–Macaulay, we show that the possible Betti sequences are (1,5,5,1), (1,5,6,2) and (1,6,8,3).

PDF

___

[1] Arslan F. Cohen-Macaulayness of tangent cones. Proceedings of the American Mathematical Society 2000; 128: 2243-2251.
[2] Arslan F, Katsabekis A, Nalbandiyan M. On the Cohen-Macaulayness of tangent cones of monomial curves in A 4 (K). Turkish Journal of Mathematics 2019; 43: 1425-1446.
[3] Arslan F, Mete P. Hilbert functions of Gorenstein monomial curves. Proceedings of the American Mathematical Society 2007; 135: 1993-2002.
[4] Bresinsky H. Symmetric semigroups of integers generated by four elements. Manuscripta Mathematica 1975; 17: 205-219.
[5] Buchsbaum D, Eisenbud D. What makes a complex exact? Journal of Algebra 1973; 323: 259-268.
[6] Herzog J, Stamate DI. On the defining equations of the tangent cone of a numerical semigroup ring. Journal of Algebra 2014; 418: 8-28.
[7] Katsabekis A. Hilbert series of tangent cones for Gorenstein monomial curves in A 4 (K). Turkish Journal of Mathematics 2021; 45: 597-616.
[8] Kunz E. The value-semigroup of a one-dimensional Gorenstein ring. Proceedings of the American Mathematical Society 1970; 25: 748-751.
[9] Mete P, Zengin EE. Minimal free resolutions of the tangent cones for Gorenstein monomial curves. Turkish Journal of Mathematics 2019; 43: 2782-2793.
[10] Stamate DI. Betti numbers for numerical semigroup rings. Multigraded Algebra and Applications, Springer Proceedings in Mathematics and Statistics 2018; 238: 133-157.
[11] Şahin M, Şahin N. Betti numbers for certain Cohen-Macaulay tangent cones. Bulletin of the Australian Mathematical Society 2019; 99: 68-77.