Abstract

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The lecture is based on joint work of the speaker with H. Derksen and J. Weyman.

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Let $\Lambda$ be a basic finite dimensional algebra over an algebraically closed 

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field $K$. We briefly review existing information about the degenerations of finite 
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dimensional representations of $\Lambda$. Roughly, our main results say that 

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substantial slices of the representations with fixed dimension vector $\bold d$ 

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are always classifiable in the most stringent geometric terms. Namely, given any 

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finite dimensional semisimple $\Lambda$-module $T$, the isomorphism classes of the

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representations with dimension vector $\bold d$ which are degeneration-maximal 

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among those with top $T$ always have a fine moduli space. Depending on a Borel 

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subgroup of the automorphism group $Aut_\Lambda(T)$, this moduli space can be 

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explicitly identified as a certain closed subvariety of the Grassmannian parametrizing 

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variety for the modules with dimension vector $\bold d$.  Despite classifiability, the 

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representation theory of the addressed class of modules is rich:  Any projective 

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variety can be realized as a fine moduli space in this context, for suitable 
$\Lambda$, 

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$\bold d$ and $T$.  We supplement the discussion of the moduli space with a 

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structural description of the modules which are degeneration-maximal among those 

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with fixed top; this characterization arises as 
a consequence of the geometric findings.