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Criteria for $\sigma$-ampleness

Author: Dennis S. Keeler
Journal: J. Amer. Math. Soc. 13 (2000), 517-532
MSC (2000): Primary 14A22, 14F17, 14J50, 16P90, 16S38, 16W50
Published electronically: March 29, 2000
MathSciNet review: 1758752
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In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a $\sigma$-ample divisor, where $\sigma$ is an automorphism of a projective scheme $X$. Many open questions regarding $\sigma$-ample divisors have remained.

We derive a relatively simple necessary and sufficient condition for a divisor on $X$ to be $\sigma$-ample. As a consequence, we show right and left $\sigma$-ampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GK-dimension. We also characterize which automorphisms $\sigma$ yield a $\sigma$-ample divisor.

References [Enhancements On Off] (What's this?)

  • [AS] M. Artin and J. T. Stafford, Noncommutative graded domains with quadratic growth, Invent. Math. 122 (1995), no. 2, 231-276. MR 96g:16027
  • [ATV] M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Vol. I, Birkhäuser Boston, Boston, MA, 1990, pp. 33-85. MR 92e:14002
  • [AV] M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, J. Algebra 133 (1990), no. 2, 249-271. MR 91k:14003
  • [AZ] M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228-287. MR 96a:14004
  • [Fj] Takao Fujita, Vanishing theorems for semipositive line bundles, Algebraic geometry (Tokyo-Kyoto, 1982), Springer, Berlin, 1983, pp. 519-528. MR 85g:14023
  • [Fl] William Fulton, Intersection theory, second ed., Springer-Verlag, Berlin, 1998. MR 99d:14003
  • [H] Robin Hartshorne, Ample subvarieties of algebraic varieties, Springer-Verlag, Berlin, 1970, Notes written in collaboration with C. Musili. Lecture Notes in Mathematics, Vol. 156. MR 44:211
  • [K] Steven L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293-344. MR 34:5834
  • [Ko] János Kollár, Rational curves on algebraic varieties, Springer-Verlag, Berlin, 1996. MR 98c:14001
  • [KL] Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, revised ed., American Mathematical Society, Providence, RI, 2000. CMP 2000:04
  • [R] Miles Reid, Canonical $3$-folds, Journées de Géométrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, pp. 273-310. MR 82i:14025
  • [SS] S. P. Smith and J. T. Stafford, Regularity of the four-dimensional Sklyanin algebra, Compositio Math. 83 (1992), no. 3, 259-289. MR 93h:16037
  • [St1] Darin R. Stephenson, Artin-Schelter regular algebras of global dimension three, J. Algebra 183 (1996), no. 1, 55-73. MR 97h:16053
  • [St2] -, Algebras associated to elliptic curves, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2317-2340. MR 97m:16080
  • [St3] -, The geometry of noncommutative graded algebras, preliminary version, 1998.
  • [SZ] Darin R. Stephenson and James J. Zhang, Growth of graded Noetherian rings, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1593-1605. MR 97g:16033
  • [V] James S. Vandergraft, Spectral properties of matrices which have invariant cones, SIAM J. Appl. Math. 16 (1968), 1208-1222. MR 39:5599
  • [W] Joachim Wehler, ${K}3$-surfaces with Picard number $2$, Arch. Math. (Basel) 50 (1988), no. 1, 73-82. MR 89b:14054

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Additional Information

Dennis S. Keeler
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Keywords: Noetherian graded rings, noncommutative projective geometry, automorphisms, vanishing theorems
Received by editor(s): December 13, 1999
Published electronically: March 29, 2000
Additional Notes: The author was partially supported by NSF grant DMS-9801148.
Article copyright: © Copyright 2000 American Mathematical Society

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