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

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Abstract

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

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

Dates
First available in Project Euclid: 9 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1565337619

Digital Object Identifier
doi:10.11650/tjm/190710

Subjects
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]

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

Citation

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


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References

  • 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.