Rocky Mountain Journal of Mathematics

On the projective dimension of $5$ quadric almost complete intersections with low multiplicities

Sabine El Khoury

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $S$ be a polynomial ring over an algebraically closed field $k$ and $ \mathfrak p =(x,y,z,w) $ a homogeneous height $4$ prime ideal. We give a finite characterization of the degree $2$ component of ideals primary to $\mathfrak p$, with multiplicity $e \leq 3$. We use this result to give a tight bound on the projective dimension of almost complete intersections generated by five quadrics with $e \leq 3$.

Article information

Rocky Mountain J. Math., Volume 49, Number 5 (2019), 1491-1546.

First available in Project Euclid: 19 September 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13D02: Syzygies, resolutions, complexes
Secondary: 13D05: Homological dimension

projective dimension almost complete intersections primary ideals


Khoury, Sabine El. On the projective dimension of $5$ quadric almost complete intersections with low multiplicities. Rocky Mountain J. Math. 49 (2019), no. 5, 1491--1546. doi:10.1216/RMJ-2019-49-5-1491.

Export citation


  • T. Ananyan and M. Hochster, Small subalgebras of polynomial rings and Stillman's conjecture, preprint (2016), arXiv:1610.09268.
  • T. Ananyan and M. Hochster, Ideals generated by quadratic polynomials, Math. Res. Lett. 19 (2012), 233–244.
  • D. Eisenbud, The geometry of syzygies: a second course in commutative algebra and algebraic geometry, Graduate Texts in Math. 229, Springer (2005).
  • B. Engheta, Bounds on projective dimension, Ph.D. thesis, University of Kansas (2005).
  • B. Engheta, On the projective dimension and the unmixed part of three cubics, J. Algebra 316 (2007), 715–734.
  • B. Engheta, A bound on the projective dimension of three cubics, J. Sym. Comp. 45 (2010), 60–73.
  • G. Fløystad, J. McCullough and I. Peeva, Three themes of syzygies, Bull. Amer. Math. Soc. (N.S.) 53 (2016), 415–435.
  • D.R. Grayson and M.E. Stillman, Macaulay 2, software available at
  • J. Harris, Algebraic geometry: a first course, Graduate Texts in Math. 133, Springer (1992).
  • C. Huneke, P. Mantero, J. McCullough and A. Seceleanu, A tight bound on the projective dimension of four quadrics, J. Pure Appl. Algebra 222 (2018), 2524–2551.
  • C. Huneke, P. Mantero, J. McCullough and A. Seceleanu, Multiple structures with arbitrarily large projective dimension supported on linear subspaces, J. Algebra 447 (2016), 183–205.
  • C. Huneke, P. Mantero, J. McCullough and A. Seceleanu, The projective dimension of codimension two algebras presented by quadrics, J. Algebra 393 (2013), 170–186.
  • C. Huneke and I. Swanson, Integral closure of ideals, rings, and modules, London Math. Society Lecture Note Series 336, Cambridge University Press (2006).
  • N. Manolache, Codimension two linear varieties with nilpotent structures, Math. Zeitschrift 210 (1992), 573–580.
  • N. Manolache, Cohen-Macaulay nilpotent schemes, pp. 235–248 in Recent advances in geometry and topology (Cluj-Napoca, Romania, 2003), Cluj University Press (2004).
  • P. Mantero and J. McCullough, A finite classification of $(x,y)$-primary ideals of low multiplicity, Collect. Math. 69 (2018), 107–130.
  • P. Mantero and J. McCullough, The projective dimension of three cubics is at most 5, preprint (2018), arXiv:1801.08195.
  • J. McCullough and A. Seceleanu, Bounding projective dimension, pp. 551–576 in Commutative algebra, Springer (2013).
  • M. Nagata, Local rings, Interscience, New York (1962). Reprinted Krieger, Huntington, NY (1975).
  • I. Peeva and M. Stillman, Open problems on syzygies and Hilbert functions, J. Commut. Algebra 1 (2009), 159–195.
  • C. Peskine and L. Szpiro, Liaison des variétés algébriques, Invent. Math. 26 (1974), 271–302.
  • P. Samuel, La notion de multiplicité en algèbre et en géométrie algébrique, I–II, J. Math. Pures. Appl. $(9)$ 30 (1951), 159–274.
  • E. Tavanfar, Annihilators of Koszul homologies and almost complete intersections, preprint (2017), arXiv.1702.01111, to appear in J. Commut. Algebra.
  • J. Vatne, Multiple structures and Hartshorne's conjecture, Comm. Algebra 37 (2009), 3861–3873.