×

A theorem about Cremona maps and symbolic Rees algebras. (English) Zbl 1314.13008

The main goal of the paper under review, is to understand the structure of the symbolic Rees algebra of the base ideal of a Cremona map. The main theorem (Theorem 2.1) shows that under mild assumptions on the base ideal \(I\) and its Rees algebra \(\mathcal{R}(I)\), we have an explicit description of the symbolic Rees algebra \(\mathcal{R}^{(I)}\) and its presentation ideal in terms of that of \(\mathcal{R}(I)\). This unifies the results of A. Simis et al. [Am. J. Math. 115, No. 1, 47–75 (1993; Zbl 0791.13007)], Z. Ramos and A. Simis [J. Algebra 413, 153–197 (2014; Zbl 1311.13007), J. Algebra Appl. 14, No. 3, Article ID 1550031, 12 p. (2015; Zbl 1315.13015)]. Many interesting examples are illustrated. There is an appendix containing a proof of the Cohen-Macaulayness of the Rees algebra of the base ideal of certain plane Cremona maps of degree \(4\) that are not de Jonquières.

MSC:

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13D02 Syzygies, resolutions, complexes and commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14E05 Rational and birational maps
14E07 Birational automorphisms, Cremona group and generalizations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.1007/b82933 · Zbl 0991.14008
[2] DOI: 10.1016/j.aim.2008.03.025 · Zbl 1144.14009
[3] DOI: 10.1016/j.jpaa.2011.06.007 · Zbl 1239.13006
[4] DOI: 10.4310/MRL.2013.v20.n4.a3 · Zbl 1299.14014
[5] DOI: 10.1016/j.aim.2011.12.005 · Zbl 1251.14007
[6] DOI: 10.1016/j.jalgebra.2012.08.022 · Zbl 1275.13019
[7] DOI: 10.1016/j.aim.2006.06.007 · Zbl 1112.13006
[8] J. Herzog, A. Simis and W. V. Vasconcelos, Commutative Algebra, Lecture Notes in Pure and Applied Mathematics 84, eds. S. Greco and G. Valla (Marcel-Dekker, New York, 1983) pp. 79–169.
[9] Huneke C., London Mathematical Society Lecture Note Series 336, in: Integral Closure of Ideals, Rings, and Modules (2006) · Zbl 1117.13001
[10] DOI: 10.1016/j.jalgebra.2014.05.012 · Zbl 1311.13007
[11] DOI: 10.1142/S0219498815500310 · Zbl 1315.13015
[12] DOI: 10.1023/A:1017572213947 · Zbl 1036.14005
[13] DOI: 10.1112/S0025579300002539 · Zbl 0186.26401
[14] DOI: 10.1016/j.jalgebra.2004.03.025 · Zbl 1067.14014
[15] DOI: 10.2307/2374722 · Zbl 0791.13007
[16] DOI: 10.1007/BF01446637 · Zbl 0819.13003
[17] DOI: 10.1017/S0305004100004667 · Zbl 1096.13500
[18] DOI: 10.1090/S0025-5718-2011-02556-1 · Zbl 1259.14010
[19] DOI: 10.1017/CBO9780511574726
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.