Irreducible components of module varieties: projective equations and rationality

2011/08/14 by Huisgen-Zimmermann, B., Goodearl, K. R.
#16G10 #16G20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1108.2833

Abstract

We expand the existing arsenal of methods for exploring the irreducible components of the varieties Rep(A,\bold d) which parametrize the representations with dimension vector \bold d of a finite dimensional algebra A. To do so, we move back and forth between Rep(A,\bold d) and a projective variety, GRASS(A,\bold d), parametrizing the same set of isomorphism classes of modules. In particular, we show the irreducible components to be accessible in a highly compressed format within the projective setting. Our results include necessary and sufficient conditions for unirationality, smoothness, and normality, followed by applications. Moreover, they provide equational access to the irreducible components of GRASS(A,\bold d) and techniques for deriving qualitative information regarding both the affine and projective scenarios.

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