A family of reflexive vector bundles of reduction number one
DOI:
https://doi.org/10.7146/math.scand.a-111889Abstract
A difficult issue in modern commutative algebra asks for examples of modules (more interestingly, reflexive vector bundles) having prescribed reduction number $r\geq 1$. The problem is even subtler if in addition we are interested in good properties for the Rees algebra. In this note we consider the case $r=1$. Precisely, we show 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.
References
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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
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
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
Vasconcelos, W. V., Integral closure: Rees algebras, multiplicities, algorithms, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.
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
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
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
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
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.
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
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
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.
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.
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
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.
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
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
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
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
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.
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
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
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
Vasconcelos, W. V., Integral closure: Rees algebras, multiplicities, algorithms, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.
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Published
2019-06-17
How to Cite
Miranda-Neto, C. B. (2019). A family of reflexive vector bundles of reduction number one. MATHEMATICA SCANDINAVICA, 124(2), 188–202. https://doi.org/10.7146/math.scand.a-111889
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