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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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GK-dimension of birationally commutative surfaces
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by D. Rogalski PDF
Trans. Amer. Math. Soc. 361 (2009), 5921-5945 Request permission


Let $k$ be an algebraically closed field, let $K/k$ be a finitely generated field extension of transcendence degree $2$, let $\sigma \in \operatorname {Aut}_k(K)$, and let $A \subseteq Q = K[t; \sigma ]$ be an $\mathbb {N}$-graded subalgebra with $\dim _k A_n < \infty$ for all $n \geq 0$. Then if $A$ is big enough in $Q$ in an appropriate sense, we prove that $\operatorname {GK} A = 3,4,5,$ or $\infty$, with the exact value depending only on the geometric properties of $\sigma$. The proof uses techniques in the birational geometry of surfaces which are of independent interest.
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Additional Information
  • D. Rogalski
  • Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112
  • MR Author ID: 734142
  • Email:
  • Received by editor(s): September 19, 2007
  • Published electronically: June 15, 2009
  • Additional Notes: The author was partially supported by the NSF through grants DMS-0202479 and DMS-0600834.
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 5921-5945
  • MSC (2000): Primary 14A22, 14E05, 16P90, 16S38, 16W50
  • DOI:
  • MathSciNet review: 2529919