Normal cones of monomial primes

Hübl, Reinhold; Swanson, Irena

doi:10.1090/S0025-5718-02-01416-3

Normal cones of monomial primes
HTML articles powered by AMS MathViewer

by Reinhold Hübl and Irena Swanson PDF

Math. Comp. 72 (2003), 459-475 Request permission

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.

References

Teresa Cortadellas and Santiago Zarzuela, On the depth of the fiber cone of filtrations, J. Algebra 198 (1997), no. 2, 428–445. MR 1489906, DOI 10.1006/jabr.1997.7147
R. C. Cowsik and M. V. Nori, On the fibres of blowing up, J. Indian Math. Soc. (N.S.) 40 (1976), no. 1-4, 217–222 (1977). MR 572990
David Eisenbud and Barry Mazur, Evolutions, symbolic squares, and Fitting ideals, J. Reine Angew. Math. 488 (1997), 189–201. MR 1465370
Patrizia Gianni, Barry Trager, and Gail Zacharias, Gröbner bases and primary decomposition of polynomial ideals, J. Symbolic Comput. 6 (1988), no. 2-3, 149–167. Computational aspects of commutative algebra. MR 988410, DOI 10.1016/S0747-7171(88)80040-3
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.
M. Herrmann, S. Ikeda, and U. Orbanz, Equimultiplicity and blowing up, Springer-Verlag, Berlin, 1988. An algebraic study; With an appendix by B. Moonen. MR 954831, DOI 10.1007/978-3-642-61349-4
Reinhold Hübl, Evolutions and valuations associated to an ideal, J. Reine Angew. Math. 517 (1999), 81–101. MR 1728546, DOI 10.1515/crll.1999.099
R. Hübl and C. Huneke, Fiber cones and the integral closure of ideals, Collect. Math., 52 (2001), 85-100.
R. Hübl and I. Swanson, Discrete valuations centered on local domains, Jour. Pure Appl. Algebra, 161 (2001), 145-166.
Craig Huneke, The theory of $d$-sequences and powers of ideals, Adv. in Math. 46 (1982), no. 3, 249–279. MR 683201, DOI 10.1016/0001-8708(82)90045-7
Craig Huneke and Judith D. Sally, Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988), no. 2, 481–500. MR 943272, DOI 10.1016/0021-8693(88)90274-8
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
Marcel Morales and Aron Simis, Symbolic powers of monomial curves in $\textbf {P}^3$ lying on a quadric surface, Comm. Algebra 20 (1992), no. 4, 1109–1121. MR 1154405, DOI 10.1080/00927879208824394
Dilip P. Patil, Minimal sets of generators for the relation ideals of certain monomial curves, Manuscripta Math. 80 (1993), no. 3, 239–248. MR 1240646, DOI 10.1007/BF03026549
Dilip P. Patil and Balwant Singh, Generators for the derivation modules and the relation ideals of certain curves, Manuscripta Math. 68 (1990), no. 3, 327–335. MR 1065934, DOI 10.1007/BF02568767
Kishor Shah, On the Cohen-Macaulayness of the fiber cone of an ideal, J. Algebra 143 (1991), no. 1, 156–172. MR 1128652, DOI 10.1016/0021-8693(91)90257-9
Kishor Shah, On equimultiple ideals, Math. Z. 215 (1994), no. 1, 13–24. MR 1254811, DOI 10.1007/BF02571697
Wolmer V. Vasconcelos, Arithmetic of blowup algebras, London Mathematical Society Lecture Note Series, vol. 195, Cambridge University Press, Cambridge, 1994. MR 1275840, DOI 10.1017/CBO9780511574726