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Normal cones of monomial primes

Authors: Reinhold Hübl and Irena Swanson
Journal: Math. Comp. 72 (2003), 459-475
MSC (2000): Primary 13-04, 13C14
Published electronically: June 6, 2002
MathSciNet review: 1933831
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Abstract: We explicitly calculate the normal cones of all monomial primes which define the curves of the form $(t^{L}, t^{L+1}, \ldots , t^{L+n})$, where $n \le 4$. All of these normal cones are reduced and Cohen-Macaulay, and their reduction numbers are independent of the reduction. These monomial primes are new examples of integrally closed ideals for which the product with the maximal homogeneous ideal is also integrally closed.

Substantial use was made of the computer algebra packages Maple and Macaulay2.

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  • [CZ] T. Cortadellas and S. Zarzuela, On the depth of the fiber cone of filtrations, J. Algebra, 198 (1997), 428-445. MR 98k:13005
  • [CN] R. Cowsik and M. Nori, On the fibers of blowing-up, J. Indian Math. Soc., 40 (1976), 217-222. MR 58:28011
  • [EM] D. Eisenbud and B. Mazur, Evolutions, symbolic squares and Fitting ideals, Jour. Reine Angew. Math., 488 (1997), 189-201. MR 98h:13035
  • [GTZ] P. Gianni, B. Trager and G. Zacharias, Gröbner bases and primary decompositions of polynomial ideals, J. Symbolic Comput., 6 (1988), 149-167. MR 90f:68091
  • [G] P. Gimenez, ``Étude de la fibre spéciale de l'éclatement d'une varieté monomiale en codimension deux'', Thèse de Doctorat de Mathématiques de l'Université Joseph Fourier, Grenoble, 1993.
  • [HIO] M. Herrmann, S. Ikeda and U. Orbanz, Equimultiplicity and Blowing up, Springer-Verlag, 1988. MR 89g:13012
  • [H] R. Hübl, Evolutions and valuations associated to an ideal, Jour. Reine Angew. Math., 517 (1999), 81-101. MR 2000j:13047
  • [HH] R. Hübl and C. Huneke, Fiber cones and the integral closure of ideals, Collect. Math., 52 (2001), 85-100.
  • [HS] R. Hübl and I. Swanson, Discrete valuations centered on local domains, Jour. Pure Appl. Algebra, 161 (2001), 145-166.
  • [Hu] C. Huneke, The theory of d-sequences and powers of ideals, Adv. in Math., 46 (1982), 249-279. MR 84g:13021
  • [HSa] C. Huneke and J. Sally, Birational extensions in dimension two and integrally closed ideals, J. Algebra, 115 (1988), 481-500. MR 89e:13025
  • [M] H. Matsumura, Commutative Algebra, 2nd edition. Benjamin/Cummings, Reading, Ma., 1980. MR 82i:13003
  • [MS] M. Morales and A. Simis, Symbolic powers of monomial curves in $\mathbb{P}^{n}$ lying on a quadric surface, Comm. Algebra, 20 (1992), 1109-1121. MR 93c:13005
  • [P] D. P. Patil, Minimal sets of generators for the relation ideals of certain monomial curves, Manu. Math., 80 (1993), 239-248. MR 94h:14026
  • [PS] D. P. Patil and B. Singh, Generators for the derivation modules and the relation ideals of certain curves, Manu. Math., 68 (1990), 327-335. MR 91g:14019
  • [Sh1] K. Shah, On the Cohen-Macaulayness of the fiber cone of an ideal, J. Algebra, 143 (1991), 156-172. MR 92k:13014
  • [Sh2] K. Shah, On equimultiple ideals, Math. Z., 215 (1994), 13-24. MR 95j:13021
  • [Va] W. Vasconcelos, Arithmetic of Blowup Algebras, London Mathematical Society Lecture Note Series 195, Cambridge University Press, 1994. MR 95g:13005

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Additional Information

Reinhold Hübl
Affiliation: NWF I - Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Irena Swanson
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003-8001

Keywords: Monomial prime, normal cone, Cohen-Macaulay, Gorenstein
Received by editor(s): February 22, 2000
Received by editor(s) in revised form: February 28, 2001
Published electronically: June 6, 2002
Additional Notes: The first author was partially supported by a Heisenberg–Stipendium of the DFG
The second author was partially supported by the National Science Foundation.
Article copyright: © Copyright 2002 American Mathematical Society

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