Abstract: 

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We report on work in progress on the prime spectra of birationally commutative 

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algebras of low GK-dimension. We work over the complex numbers.
 
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Recall that if X is a complex projective surface, s is an automorphism of X, 
and L

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is an s-ample invertible sheaf on X, then the twisted homogeneous 
coordinate

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ring B(X, L, s) may, somewhat counterintuitively, have GK-dimension 3 or 5.

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Let R be a noetherian connected graded domain that is birational to a 
GK-5 twisted

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homogeneous coordinate ring of a projective surface X. We show 
that there are

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two dichotomous possibilities for the prime spectrum of R.
If Spec(R) is nontrivial, 

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then R has a 1-dimensional space of height 1 primes, 
all but countably many of which 

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are primitive. However, R has prime factors of 
arbitrarily large PI-degree. We also 

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show that in this case R is, up to finite 
dimension, a twisted homogeneous coordinate 

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ring itself.

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On the other hand, if Spec(R) = {0}, then R need not be a twisted homogeneous 

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coordinate ring. In this case, R has the amusing property that all of its 
subalgebras 

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are noetherian.