Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE KÄHLER DIFFERENT OF A SET OF POINTS IN ℙm × ℙn

  • Hoa, Nguyen T. (Department of Mathematics University of Education Hue University) ;
  • Linh, Tran N.K. (Department of Mathematics University of Education Hue University) ;
  • Long, Le N. (Fakultat fur Informatik und Mathematik Universitat Passau and Department of Mathematics University of Education Hue University) ;
  • Nhan, Phan T.T. (Department of Mathematics University of Education Hue University) ;
  • Nhi, Nguyen T.P. (Department of Mathematics University of Education Hue University)
  • Received : 2021.07.22
  • Accepted : 2021.10.29
  • Published : 2022.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

Given an ACM set 𝕏 of points in a multiprojective space ℙm×ℙn over a field of characteristic zero, we are interested in studying the Kähler different and the Cayley-Bacharach property for 𝕏. In ℙ1×ℙ1, the Cayley-Bacharach property agrees with the complete intersection property and it is characterized by using the Kähler different. However, this result fails to hold in ℙm×ℙn for n > 1 or m > 1. In this paper we start an investigation of the Kähler different and its Hilbert function and then prove that 𝕏 is a complete intersection of type (d1, …, dm, d'1, …, d'n) if and only if it has the Cayley-Bacharach property and the Kähler different is non-zero at a certain degree. We characterize the Cayley-Bacharach property of 𝕏 under certain assumptions.

Keywords

Acknowledgement

The authors thank Martin Kreuzer and Elena Guardo for their encouragement to elaborate some results presented here. The first author and the last two authors were supported by University of Education - Hue University under grant number T.20-TN.SV-01. The second author was partially supported by the Program for Research Activities of Senior Researchers of VAST under the grant number NVCC01.11/21-21. The second and third authors were partially supported by Hue University, Grant No. NCM.DHH.2020.15 and DHH2021-03-159. Last, but not least, we are extremely thankful to the referee for his/her very detailed and enlightening comments.

References

  1. S. Bouchiba and S. Kabbaj, Tensor products of Cohen-Macaulay rings: solution to a problem of Grothendieck, J. Algebra 252 (2002), no. 1, 65-73. https://doi.org/10.1016/S0021-8693(02)00019-4
  2. M. Chardin and N. Nemati, Multigraded regularity of complete intersections, available at arXiv:2012.14899v1 (2020).
  3. G. Favacchio and J. Migliore, Multiprojective spaces and the arithmetically Cohen-Macaulay property, Math. Proc. Cambridge Philos. Soc. 166 (2019), no. 3, 583-597. https://doi.org/10.1017/S0305004118000142
  4. A. V. Geramita, M. Kreuzer, and L. Robbiano, Cayley-Bacharach schemes and their canonical modules, Trans. Amer. Math. Soc. 339 (1993), no. 1, 163-189. https://doi.org/10.2307/2154213
  5. E. Guardo, M. Kreuzer, T. N. K. Linh, and L. N. Long, Kahler differentials for fat point schemes in ℙ1×ℙ1, J. Commut. Algebra 13 (2021), no. 2, 179-207. https://doi.org/10.1216/jca.2021.13.179
  6. E. Guardo, T. N. K. Linh, and L. N. Long, A presentation of the Kahler differential module for a fat point scheme in ℙn1×···×ℙnk, ITM Web of Conferences 20 (2018), no. 4, 01007. https://doi.org/10.1051/itmconf/20182001007
  7. E. Guardo and A. Van Tuyl, Fat points in ℙ1×ℙ1 and their Hilbert functions, Canad. J. Math. 56 (2004), no. 4, 716-741. https://doi.org/10.4153/CJM-2004-033-0
  8. E. Guardo and A. Van Tuyl, ACM sets of points in multiprojective space, Collect. Math. 59 (2008), no. 2, 191-213. https://doi.org/10.1007/BF03191367
  9. E. Guardo and A. Van Tuyl, Separators of points in a multiprojective space, Manuscripta Math. 126 (2008), no. 1, 99-13. https://doi.org/10.1007/s00229-008-0165-z
  10. E. Guardo and A. Van Tuyl, Classifying ACM sets of points in ℙ1×ℙ1 via separators, Arch. Math. (Basel) 99 (2012), no. 1, 33-36. https://doi.org/10.1007/s00013-012-0404-0
  11. E. Guardo and A. Van Tuyl, Arithmetically Cohen-Macaulay sets of points in ℙ1×ℙ1, SpringerBriefs in Mathematics, Springer, Cham, 2015.
  12. M. Kreuzer and L. N. Long, Characterizations of zero-dimensional complete intersections, Beitr. Algebra Geom. 58 (2017), no. 1, 93-129. https://doi.org/10.1007/s13366-016-0311-9
  13. M. Kreuzer, L. N. Long, and L. Robbiano, On the Cayley-Bacharach property, Comm. Algebra 47 (2019), no. 1, 328-354. https://doi.org/10.1080/00927872.2018.1476525
  14. M. Kreuzer and L. Robbiano, Computational Commutative Algebra. 1, Springer-Verlag, Berlin, 2000. https://doi.org/10.1007/978-3-540-70628-1
  15. M. Kreuzer, N. K. L. Tran, and L. N. Long, Kahler differentials and Kahler differents for fat point schemes, J. Pure Appl. Algebra 219 (2015), no. 10, 4479-4509. https://doi.org/10.1016/j.jpaa.2015.02.028
  16. E. Kunz, Kahler Differentials, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1986. https://doi.org/10.1007/978-3-663-14074-0
  17. E. Kunz, Introduction to Plane Algebraic Curves, translated from the 1991 German edition by Richard G. Belshoff, Birkhauser Boston, Inc., Boston, MA, 2005.
  18. L. Marino, A characterization of ACM 0-dimensional schemes in Q, Matematiche (Catania) 64 (2009), no. 2, 41-56.
  19. G. Scheja and U. Storch, Uber Spurfunktionen bei vollstandigen Durchschnitten, J. Reine Angew. Math. 278(279) (1975), 174-190.
  20. J. Sidman and A. Van Tuyl, Multigraded regularity: syzygies and fat points, Beitrage Algebra Geom. 47 (2006), no. 1, 67-87.
  21. The ApCoCoA Team, ApCoCoA: Applied Computations in Commutative Algebra, available at http://apcocoa.uni-passau.de.
  22. A. Van Tuyl, The border of the Hilbert function of a set of points in ℙn1×···×ℙnk, J. Pure Appl. Algebra 176 (2002), no. 2-3, 223-247. https://doi.org/10.1016/S0022-4049(02)00072-5
  23. A. Van Tuyl, The Hilbert functions of ACM sets of points in ℙn1×···×ℙnk, J. Algebra 264 (2003), no. 2, 420-441. https://doi.org/10.1016/S0021-8693(03)00232-1