Taiwanese Journal of Mathematics

An Application of Liaison Theory to Zero-dimensional Schemes

Martin Kreuzer, Tran N. K. Linh, Le Ngoc Long, and Tu Chanh Nguyen

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access


Given a $0$-dimensional scheme $\mathbb{X}$ in an $n$-dimensional projective space $\mathbb{P}^n_K$ over an arbitrary field $K$, we use liaison theory to characterize the Cayley-Bacharach property of $\mathbb{X}$. Our result extends the result for sets of $K$-rational points given in [8]. In addition, we examine and bound the Hilbert function and regularity index of the Dedekind different of $\mathbb{X}$ when $\mathbb{X}$ has the Cayley-Bacharach property.

Article information

Taiwanese J. Math., Advance publication (2019), 21 pages.

First available in Project Euclid: 9 August 2019

Permanent link to this document

Digital Object Identifier

Primary: 13C40: Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] 14M06: Linkage [See also 13C40]
Secondary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 14N05: Projective techniques [See also 51N35]

zero-dimensional scheme Cayley-Bacharach property Hilbert function liaison theory Dedekind different


Kreuzer, Martin; Linh, Tran N. K.; Long, Le Ngoc; Nguyen, Tu Chanh. An Application of Liaison Theory to Zero-dimensional Schemes. Taiwanese J. Math., advance publication, 9 August 2019. doi:10.11650/tjm/190710. https://projecteuclid.org/euclid.twjm/1565337619

Export citation


  • ApCoCoA Team, ApCoCoA: Applied Computations in Computer Algebra, available at http://apcocoa.uni-passau.de.
  • G. Bolondi, J. O. Kleppe and R. M. Miró-Roig, Maximal rank curves and singular points of the Hilbert scheme, Compositio Math. 77 (1991), no. 3, 269–291.
  • W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993.
  • K. F. E. Chong, An application of liaison theory to the Eisenbud-Green-Harris conjecture, J. Algebra 445 (2016), 221–231.
  • E. D. Davis, A. V. Geramita and F. Orecchia, Gorenstein algebras and the Cayley-Bacharach theorem, Proc. Amer. Math. Soc. 93 (1985), no. 4, 593–597.
  • G. Favacchio, E. Guardo and J. Migliore, On the arithmetically Cohen-Macaulay property for sets of points in multiprojective spaces, Proc. Amer. Math. Soc. 146 (2018), no. 7, 2811–2825.
  • L. Fouli, C. Polini and B. Ulrich, Annihilators of graded components of the canonical module, and the core of standard graded algebras, Trans. Amer. Math. Soc. 362 (2010), no. 12, 6183–6203.
  • 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.
  • L. Gold, J. Little and H. Schenck, Cayley-Bacharach and evaluation codes on complete intersections, J. Pure Appl. Algebra 196 (2005), no. 1, 91–99.
  • E. Gorla, J. C. Migliore and U. Nagel, Gröbner bases via linkage, J. Algebra 384 (2013), 110–134.
  • S. Goto and K. Watanabe, On graded rings I, J. Math. Soc. Japan 30 (1978), no. 2, 179–213.
  • E. Guardo, Schemi di “Fat Points", Ph.D. Thesis, Università di Messina, 2000.
  • J. O. Kleppe, J. C. Migliore, R. Miró-Roig, U. Nagel and C. Peterson, Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. Amer. Math. Soc. 154 (2001), no. 732, 116 pp.
  • M. Kreuzer, On the canonical module of a $0$-dimensional scheme, Canad. J. Math. 46 (1994), no. 2, 357–379.
  • M. Kreuzer, T. N. K. Linh and L. N. Long, The Dedekind different of a Cayley-Bacharach scheme, J. Algebra Appl. 18 (2019), no. 2, 1950027, 33 pp.
  • M. Kreuzer and L. N. Long, Characterizations of zero-dimensional complete intersections, Beitr. Algebra Geom. 58 (2017), no. 1, 93–129.
  • M. Kreuzer, L. N. Long and L. Robbiano, On the Cayley-Bacharach property, Comm. Algebra 47 (2019), no. 1, 328–354.
  • M. Kreuzer and L. Robbiano, On maximal Cayley-Bacharach schemes, Comm. Algebra 23 (1995), no. 9, 3357–3378.
  • ––––, Computational Commutative Algebra 2, Springer-Verlag, Heidelberg 2005.
  • ––––, Computational Linear and Commutative Algebra, Springer, Switzerland, 2016.
  • T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics 189, Springer-Verlag, New York, 1991.
  • L. N. Long, Various differents for $0$-dimensional schemes and appplications, Universtät Passau, Passau, 2015.
  • R. Maggioni and A. Ragusa, The Hilbert function of generic plane sections of curves of $\mathbb{P}^3$, Invent. Math. 91 (1988), no. 2, 253–258.
  • J. C. Migliore, Introduction to Liaison Theory and Deficiency Modules, Progress in Mathematics 165, Birkhäuser Boston, Boston, MA, 1998.
  • J. Migliore and U. Nagel, Monomial ideals and the Gorenstein liaison class of a complete intersection, Compositio Math. 133 (2002), no. 1, 25–36.
  • S. Popescu and K. Ranestad, Surfaces of degree $10$ in the projective fourspace via linear systems and linkage, J. Algebraic Geom. 5 (1996), no. 1, 13–76.