Abstract 

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Ruled orders appear in the non-commutative minimal model program for the classification 

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of orders on surfaces. They are the non-commutative analogues of ruled surfaces in this 

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setting. These non-commutative surfaces have not been studied much. In this talk, we will 

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examine various ways in which these ruled orders are indeed ruled. In particular, we will 

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see that they have lots of rational curves and even describe their line scheme.