×

zbMATH — the first resource for mathematics

A family of reflexive vector bundles of reduction number one. (English) Zbl 1423.13045
Modules (reflexive vector bundles) having prescribed reduction number \(r\geq 1\) and good properties for the Rees algebra are studied. In particular, the case \(r=1\) is considered. The author shows that the module of logarithmic vector fields of the Fermat divisor of any degree in projective 3-space is a reflexive vector bundle of reduction number 1 and Gorenstein Rees ring.
MSC:
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13C14 Cohen-Macaulay modules
13C40 Linkage, complete intersections and determinantal ideals
Software:
Macaulay2
PDF BibTeX XML Cite
Full Text: DOI
References:
[2] Bayer, D. and Stillman, M., Macaulay: A system for computation in algebraic geometry and commutative algebra, available via anonymous ftp from math.harvard.edu, 1992.
[3] Corso, A., Ghezzi, L., Polini, C., and Ulrich, B., Cohen-Macaulayness of special fiber rings, Comm. Algebra 31 (2003), no. 8, 3713–3734. https://doi.org/10.1081/AGB-120022439 · Zbl 1057.13007
[4] Corso, A. and Polini, C., Links of prime ideals and their Rees algebras, J. Algebra 178 (1995), no. 1, 224–238. https://doi.org/10.1006/jabr.1995.1346 · Zbl 0848.13015
[5] Corso, A., Polini, C., and Vasconcelos, W. V., Links of prime ideals, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 3, 431–436. https://doi.org/10.1017/S0305004100072212 · Zbl 0811.13003
[6] Corso, A., Polini, C., and Vasconcelos, W. V., Multiplicity of the special fiber of blowups, Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 2, 207–219. https://doi.org/10.1017/S0305004105009023 · Zbl 1098.13030
[7] Cortadellas, T. and Zarzuela, S., On the Cohen-Macaulay property of the fiber cone of ideals with reduction number at most one, in “Commutative algebra, algebraic geometry, and computational methods (Hanoi, 1996)”, Springer, Singapore, 1999, pp. 215–222. · Zbl 0945.13016
[8] D’Cruz, C. and Verma, J. K., Hilbert series of fiber cones of ideals with almost minimal mixed multiplicity, J. Algebra 251 (2002), no. 1, 98–109. https://doi.org/10.1006/jabr.2001.9139
[9] Eisenbud, D., Huneke, C., and Ulrich, B., What is the Rees algebra of a module?, Proc. Amer. Math. Soc. 131 (2003), no. 3, 701–708. https://doi.org/10.1090/S0002-9939-02-06575-9 · Zbl 1038.13002
[10] Grayson, D. R. and Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/, 2008.
[11] Hayasaka, F., Modules of reduction number one, in “The Second Japan-Vietnam Joint Seminar on Commutative Algebra, March 20–25, 2006” (Goto, S., ed.), Meiji Institute for Mathematical Sciences, 2006, pp. 51–60.
[12] Huneke, C. and Sally, J. D., Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988), no. 2, 481–500. https://doi.org/10.1016/0021-8693(88)90274-8 · Zbl 0658.13017
[13] Huneke, C. and Swanson, I., Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. · Zbl 1117.13001
[14] Jayanthan, A. V. and Verma, J. K., Fiber cones of ideals with almost minimal multiplicity, Nagoya Math. J. 177 (2005), 155–179. https://doi.org/10.1017/S0027763000009089 · Zbl 1075.13011
[15] Katz, D. and Kodiyalam, V., Symmetric powers of complete modules over a two-dimensional regular local ring, Trans. Amer. Math. Soc. 349 (1997), no. 2, 747–762. https://doi.org/10.1090/S0002-9947-97-01819-9 · Zbl 0864.13003
[16] Miranda-Neto, C. B., Graded derivation modules and algebraic free divisors, J. Pure Appl. Algebra 219 (2015), no. 12, 5442–5466. https://doi.org/10.1016/j.jpaa.2015.05.026 · Zbl 1327.13096
[17] Miranda-Neto, C. B., A module-theoretic characterization of algebraic hypersurfaces, Canad. Math. Bull. 61 (2018), no. 1, 166–173. https://doi.org/10.4153/CMB-2016-099-6 · Zbl 1403.14079
[18] Saito, K., Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 265–291. · Zbl 0496.32007
[19] Shah, K., On the Cohen-Macaulayness of the fiber cone of an ideal, J. Algebra 143 (1991), no. 1, 156–172. https://doi.org/10.1016/0021-8693(91)90257-9 · Zbl 0752.13004
[20] Simis, A., Ulrich, B., and Vasconcelos, W. V., Rees algebras of modules, Proc. London Math. Soc. (3) 87 (2003), no. 3, 610–646. https://doi.org/10.1112/S0024611502014144 · Zbl 1099.13008
[21] Vasconcelos, W. V., Arithmetic of blowup algebras, London Mathematical Society Lecture Note Series, vol. 195, Cambridge University Press, Cambridge, 1994. https://doi.org/10.1017/CBO9780511574726 · Zbl 0813.13008
[22] Vasconcelos, W. V., Integral closure: Rees algebras, multiplicities, algorithms, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. · Zbl 1082.13006
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.