## GK-dimension of birationally commutative surfaces

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## Abstract:

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.## References

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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: drogalsk@math.ucsd.edu
- 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: https://doi.org/10.1090/S0002-9947-09-04885-5
- MathSciNet review: 2529919