GKdimension of birationally commutative surfaces
Author:
D. Rogalski
Journal:
Trans. Amer. Math. Soc. 361 (2009), 59215945
MSC (2000):
Primary 14A22, 14E05, 16P90, 16S38, 16W50
Published electronically:
June 15, 2009
MathSciNet review:
2529919
Fulltext PDF
Abstract 
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Abstract: Let be an algebraically closed field, let be a finitely generated field extension of transcendence degree , let , and let be an graded subalgebra with for all . Then if is big enough in in an appropriate sense, we prove that or , with the exact value depending only on the geometric properties of . The proof uses techniques in the birational geometry of surfaces which are of independent interest.
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 M. Artin and J. T. Stafford, Noncommutative graded domains with quadratic growth, Invent. Math. 122 (1995), no. 2, 231276. MR 96g:16027
 [AV]
 M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, J. Algebra 133 (1990), no. 2, 249271. MR 91k:14003
 [BPV]
 W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, SpringerVerlag, Berlin, 1984. MR 749574 (86c:32026)
 [DF]
 J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), no. 6, 11351169. MR 2002k:32028
 [Fj]
 Takao Fujita, Vanishing theorems for semipositive line bundles, Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 519528. MR 85g:14023
 [Ha]
 Robin Hartshorne, Algebraic geometry, SpringerVerlag, New York, 1977, Graduate Texts in Mathematics, No. 52. MR 57:3116
 [Ke1]
 D. S. Keeler, Criteria for ampleness, J. Amer. Math. Soc. 13 (2000), no. 3, 517532. MR 2001d:14003
 [Ke2]
 D. S. Keeler, Ample filters and Frobenius amplitude, preprint, arXiv:math/0603388.
 [KL]
 Günter R. Krause and Thomas H. Lenagan, Growth of algebras and GelfandKirillov dimension, revised ed., American Mathematical Society, Providence, RI, 2000. MR 2000j:16035
 [KRS]
 D. S. Keeler, D. Rogalski, and J. T. Stafford, Naïve noncommutative blowing up, Duke Math. J. 126 (2005), no. 3, 491546. MR 2120116
 [La]
 Robert Lazarsfeld, Positivity in algebraic geometry I, SpringerVerlag, Berlin, 2004. MR 2095471 (2005k:14001a)
 [RS1]
 D. Rogalski and J. T. Stafford, A class of noncommutative projective surfaces, preprint, arXiv:math/0612657.
 [RS2]
 D. Rogalski and J. T. Stafford, Naïve noncommutative blowups at zerodimensional schemes, J. Algebra 318 (2007), no. 2, 794833. MR 2371973
 [RZ]
 D. Rogalski and J. J. Zhang, Canonical maps to twisted rings, Math. Z. 259 (2008), no. 2, 433455. MR 2390090
 [Sh]
 Igor R. Shafarevich, Basic algebraic geometry I, second ed., SpringerVerlag, Berlin, 1994. MR 1328833 (95m:14001)
 [V]
 James S. Vandergraft, Spectral properties of matrices which have invariant cones, SIAM J. Appl. Math. 16 (1968), 12081222. MR 0244284 (39:5599)
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Additional Information
D. Rogalski
Affiliation:
Department of Mathematics, University of California, San Diego, La Jolla, California 920930112
Email:
drogalsk@math.ucsd.edu
DOI:
http://dx.doi.org/10.1090/S0002994709048855
PII:
S 00029947(09)048855
Keywords:
GKdimension,
graded rings,
noncommutative projective geometry,
noncommutative surfaces,
birational geometry
Received by editor(s):
September 19, 2007
Published electronically:
June 15, 2009
Additional Notes:
The author was partially supported by the NSF through grants DMS0202479 and DMS0600834.
Article copyright:
© Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
