Abstract

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Let A be a domain of GK dimension in [2,3).  We show that if A has a nonzero 

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locally nilpotent derivation then the GK dimension of A is exactly 2 and either 
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A satisfies a polynomial identity or A is birationally isomorphic to a ring of 

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differential operators on a smooth affine curve.  In addition to this, we propose 

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some new invariants for the study of quotient division rings of domains of GK 

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dimension two and consider the following problem.  If D and E denote respectively 

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the quotient division rings of rings of differential operators over smooth curves 

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X and Y and if D embeds in E, does it follow that the genus of X is less than or 

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equal to the genus of Y?  Joint with Agata Smoktunowicz.