×

Normal cones of monomial primes. (English) Zbl 1038.13008

This paper studies the defining prime ideals of monomial curves of type \((t^L,t^{L+1},\dots,t^{L+n})\), where \(n \leq 4\). The main result explicitly computes their normal cones and shows that these normal cones are reduced and Cohen-Macaulay. Moreover, each normal cone is essentially determined by the residue class of \(L\) modulo \(n\). As a consequence, the reduction number of the defining prime ideal is independent of the choice of the minimal reduction and its product with the maximal ideal is integrally closed. If the base field has characteristic zero, this implies that these monomial curves are evolutionary stable.

MSC:

13C14 Cohen-Macaulay modules
14H99 Curves in algebraic geometry
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
13A15 Ideals and multiplicative ideal theory in commutative rings

Software:

Maple
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Teresa Cortadellas and Santiago Zarzuela, On the depth of the fiber cone of filtrations, J. Algebra 198 (1997), no. 2, 428 – 445. · Zbl 0898.13002 · doi:10.1006/jabr.1997.7147
[2] 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). · Zbl 0437.14028
[3] David Eisenbud and Barry Mazur, Evolutions, symbolic squares, and Fitting ideals, J. Reine Angew. Math. 488 (1997), 189 – 201. · Zbl 0912.13010
[4] 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. · Zbl 0667.13008 · doi:10.1016/S0747-7171(88)80040-3
[5] 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.
[6] M. Herrmann, S. Ikeda, and U. Orbanz, Equimultiplicity and blowing up, Springer-Verlag, Berlin, 1988. An algebraic study; With an appendix by B. Moonen. · Zbl 0649.13011
[7] Reinhold Hübl, Evolutions and valuations associated to an ideal, J. Reine Angew. Math. 517 (1999), 81 – 101. · Zbl 0945.13014 · doi:10.1515/crll.1999.099
[8] R. Hübl and C. Huneke, Fiber cones and the integral closure of ideals, Collect. Math., 52 (2001), 85-100. · Zbl 0980.13006
[9] R. Hübl and I. Swanson, Discrete valuations centered on local domains, Jour. Pure Appl. Algebra, 161 (2001), 145-166. · Zbl 1094.13502
[10] Craig Huneke, The theory of \?-sequences and powers of ideals, Adv. in Math. 46 (1982), no. 3, 249 – 279. · Zbl 0505.13004 · doi:10.1016/0001-8708(82)90045-7
[11] Craig Huneke and Judith D. Sally, Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988), no. 2, 481 – 500. · Zbl 0658.13017 · doi:10.1016/0021-8693(88)90274-8
[12] Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. · Zbl 0441.13001
[13] Marcel Morales and Aron Simis, Symbolic powers of monomial curves in \?³ lying on a quadric surface, Comm. Algebra 20 (1992), no. 4, 1109 – 1121. · Zbl 0765.14018 · doi:10.1080/00927879208824394
[14] Dilip P. Patil, Minimal sets of generators for the relation ideals of certain monomial curves, Manuscripta Math. 80 (1993), no. 3, 239 – 248. · Zbl 0805.14015 · doi:10.1007/BF03026549
[15] 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. · Zbl 0715.14026 · doi:10.1007/BF02568767
[16] Kishor Shah, On the Cohen-Macaulayness of the fiber cone of an ideal, J. Algebra 143 (1991), no. 1, 156 – 172. · Zbl 0752.13004 · doi:10.1016/0021-8693(91)90257-9
[17] Kishor Shah, On equimultiple ideals, Math. Z. 215 (1994), no. 1, 13 – 24. · Zbl 0811.13017 · doi:10.1007/BF02571697
[18] Wolmer V. Vasconcelos, Arithmetic of blowup algebras, London Mathematical Society Lecture Note Series, vol. 195, Cambridge University Press, Cambridge, 1994. · Zbl 0813.13008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.