Research Article
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Explicit Free Resolutions of Tangent Cones in Affine 4-Space

Year 2026, Volume: 47 Issue: 1, 151 - 155, 27.02.2026

Özgür İnce

https://doi.org/10.17776/csj.1859728
https://izlik.org/JA99ME97MN

Abstract

This paper is devoted to the study of minimal free resolutions of tangent cones associated with complete intersection monomial curves in affine 4-space. By applying the gluing technique for numerical semigroups, we characterize specific families of these curves where the tangent cone retains the complete intersection property. Our analysis focuses particularly on curves defined by 4-generated numerical semigroups, constructed by gluing two semigroups generated by two elements. Furthermore, we present explicit computations of their minimal free resolutions under specific conditions.

Keywords

Complete intersection Free resolution Gluing Monomial curve Tangent cone

Thanks

Authors would like to thank Prof. Dr. Sefa Feza Arslan for his contributions.

References

  • [1] Herzog, J., Rossi, M. E., & Valla, G. (1986). On the depth of the symmetric algebra. Transactions of the American Mathematical Society, 296, 577–606. https://doi.org/10.1090/S0002-9947-1986-0846598-8
  • [2] Vu, T. (2014). Periodicity of Betti numbers of monomial curves. Journal of Algebra, 418, 66–90. https://doi.org/10.1016/j.jalgebra.2014.07.007
  • [3] Stamate, D. I. (2018). Betti numbers for numerical semigroup rings. In Multigraded Algebra and Applications (NSA 24, pp. 133–157). Springer International Publishing. https://doi.org/10.1007/978-3-319-90493-1_8
  • [4] Herzog, J. (1970). Generators and relations of abelian semigroup rings. Manuscripta Mathematica, 3, 175–193. https://doi.org/10.1007/BF01273309
  • [5] Herzog, J. (1975). When is a regular sequence super regular? Nagoya Mathematical Journal, 83, 183–195. https://doi.org/10.1017/S0027763000019474
  • [6] Robbiano, L., & Valla, G. (1980). On the equations defining tangent cones. Mathematical Proceedings of the Cambridge Philosophical Society, 88(2), 281–297. https://doi.org/10.1017/S0305004100057583
  • [7] Arslan, F. (1999). Cohen-Macaulayness of tangent cones. Proceedings of the American Mathematical Society, 128, 2243–2251.https://doi.org/10.1090/S0002-9939-99-05229-6
  • [8] Bresinsky, H. (1975). On prime ideals with generic zero x_i=t^(n_i ). Proc. Amer. Math. Soc., 47(2) 329–332. https://www.ams.org/journals/proc/1975-047-02/S0002-9939-1975-0389912-0/
  • [9] Barucci, V., Fröberg, R., & Şahin, M. (2014). On free resolutions of some semigroup rings. Journal of Pure and Applied Algebra, 218, 1107–1116.https://doi.org/10.1016/j.jpaa.2013.11.007
  • [10] Şahin, M., & Şahin, N. (2019). Betti numbers for certain Cohen-Macaulay tangent cones. Bulletin of the Australian Mathematical Society, 99(1), 68–77. https://doi.org/10.1017/S0004972718000898
  • [11] Şahin, N. (2023). Free resolutions for the tangent cones of some homogeneous pseudo symmetric monomial curves. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(1), 129–136. https://doi.org/10.31801/cfsuasmas.1117855
  • [12] Mete, P., & Zengin, E. (2019). Minimal free resolutions of the tangent cones of Gorenstein monomial curves. Turkish Journal of Mathematics, 43, 2782–2793.https://doi.org/10.3906/mat-1903-15
  • [13] Mete, P. (2021). On the Betti numbers of the tangent cones for Gorenstein monomial curves. Turkish Journal of Mathematics, 45(6), 2603–2613. https://journals.tubitak.gov.tr/math/vol45/iss6/15/
  • [14]Delorme, C. (1976). Sous-monoïdes d’intersection complète de N. Annales Scientifiques de l'École Normale Supérieure, 9(1), 145–154.https://doi.org/10.24033/asens.1307
  • [15] Gimenez, P., & Srinivasan, H. (2019). The structure of the minimal free resolution of semigroup rings obtained by gluing. Journal of Pure and Applied Algebra, 223, 1411–1426. https://doi.org/10.1016/j.jpaa.2018.06.010
  • [16]Şahin, M., & Stella, L. G. (2019). Gluing semigroups and strongly indispensable free resolutions. International Journal of Algebra and Computation, 29(2), 263–278. https://doi.org/10.1142/S0218196719500012
  • [17]Arslan, F., Mete, P., & Şahin, M. (2009). Gluing and Hilbert functions of monomial curves. Proceedings of the American Mathematical Society, 137(7), 2225–2232. https://doi.org/10.1090/S0002-9939-08-09785-2
  • [18]The GAP Group. (2022). GAP – Groups, Algorithms, and Programming, Version 4.12.2 [Computer software]. https://www.gap-system.org
  • [19]Grayson, D. R., & Stillman, M. E. (n.d.). Macaulay2, a software system for research in algebraic geometry [Computer software].http://www.math.uiuc.edu/Macaulay2/
  • [20]Decker, W., Greuel, G.-M., Pfister, G., & Schönemann, H. (2022). SINGULAR 4-3-0 — A computer algebra system for polynomial computations [Computer software]. https://www.singular.uni-kl.de
  • [21]Buchsbaum, D. A., & Eisenbud, D. (1973). What makes a complex exact? Journal of Algebra, 25, 259–268. https://doi.org/10.1016/0021-8693(73)90044-6
  • [22]Arslan, F., & İnce, Ö. (2025). Complete intersection monomial curves and free resolutions. Communications in Algebra, 1–10.https://doi.org/10.1080/00927872.2025.2583313
  • [23]İnce, Ö. (2023). Free Resolutions of Tangent Cones of Complete Intersection Monomial Curves [Doctoral dissertation, Middle East Technical University].https://hdl.handle.net/11511/103041
There are 23 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Özgür İnce 0000-0002-8575-3735

Submission Date January 9, 2026
Acceptance Date February 9, 2026
Publication Date February 27, 2026
DOI https://doi.org/10.17776/csj.1859728
IZ https://izlik.org/JA99ME97MN
Published in Issue Year 2026 Volume: 47 Issue: 1

Cite

APA İnce, Ö. (2026). Explicit Free Resolutions of Tangent Cones in Affine 4-Space. Cumhuriyet Science Journal, 47(1), 151-155. https://doi.org/10.17776/csj.1859728

As of 2026, Cumhuriyet Science Journal will be published in six issues per year, released in February, April, June, August, October, and December