Abstract
Let S be a polynomial ring over an algebraically closed field k. Let x and y denote linearly independent linear forms in S so that \({\mathfrak {p}}= (x,y)\) is a height two prime ideal. This paper concerns the structure of \({\mathfrak {p}}\)-primary ideals in S. Huneke, Seceleanu, and the authors showed that for \(e \ge 3\), there are infinitely many pairwise non-isomorphic \({\mathfrak {p}}\)-primary ideals of multiplicity e. However, we show that for \(e \le 4\) there is a finite characterization of the linear, quadric and cubic generators of all such \({\mathfrak {p}}\)-primary ideals. We apply our results to improve bounds on the projective dimension of ideals generated by three cubic forms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ananyan, T., Hochster, M.: Ideals generated by quadratic polynomials. Math. Res. Lett. 19, 233–244 (2012)
Bruns, W., Herzog, J.: Cohen–Macaulay rings. In: Cambridge Stud. Adv. Math., vol. 39. Cambridge University Press, Cambridge (1993)
Eisenbud, D.: Commutative algebra. With a view toward algebraic geometry. In: Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)
Eisenbud, D., Goto, S.: Linear free resolutions and minimal multiplicity. J. Algebra 88, 89–133 (1984)
Engheta, B.: Bounds on projective dimension. Ph.D. Thesis, University of Kansas (2005)
Engheta, B.: On the projective dimension and the unmixed part of three cubics. J. Algebra 316, 715–734 (2007)
Engheta, B.: A bound on the projective dimension of three cubics. J. Symb. Comput. 45, 60–73 (2010)
Fløystad, G., McCullough, J., Peeva, I.: Three themes of syzygies. Bull. Am. Math. Soc. (N.S.) 53(3), 415–435 (2016)
Huneke, C., Ulrich, B.: The structure of linkage. Ann. Math. 126, 277–334 (1987)
Huneke, C., Mantero, P., McCullough, J., Seceleanu, A.: Multiple structures on linear varieties with high projective dimension. J. Algebra 447, 183–205 (2016)
Huneke, C., Mantero, P., McCullough, J., Seceleanu, A.: A tight bound on the projective dimension of 4 quadrics. J. Pure Appl. Algebra (to appear)
Macaulay 2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
Manolache, N.: Codimension two linear varieties with nilpotent structures. Math. Z. 210(4), 573–580 (1992)
Manolache, N.: Cohen–Macaulay nilpotent schemes. In: Andrica, D., Blaga, P.A. (eds.) Recent Advances in Geometry and Topology, pp. 235–248. Cluj University Press, Cluj-Napoca (2004)
McCullough, J., Seceleanu, A.: Bounding projective dimension. In: Peeva, I. (ed.) Commutative Algebra. Expository papers dedicated to David Eisenbud on the occasion of his 65th birthday. Springer, New York (2013)
Peeva, I., Stillman, M.: Open problems on syzygies and Hilbert functions. J. Commut. Algebra 1(1), 159–195 (2009)
Peskine, C., Szpiro, L.: Liaison des variétés algebriques. Invent. Math. 26, 271–302 (1974)
Vatne, J.: Multiple structures and Hartshorne’s conjecture. Commun. Algebra 37(11), 3861–3873 (2009)
Acknowledgements
The authors thank the referee for finding several errors and helping to greatly improve this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mantero, P., McCullough, J. A finite classification of (x, y)-primary ideals of low multiplicity. Collect. Math. 69, 107–130 (2018). https://doi.org/10.1007/s13348-017-0196-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-017-0196-4