Irreducible components of module varieties: projective equations and rationality

Details Irreducible components of module varieties: projective equations and rationality Irreducible components of module varieties: projective equations and rationality

2011-08-14

arxiv.org/abs/1108.2833v2

Mathematics

Representation Theory

27 pages; 9 diagrams in xypic. Section 5 slightly shortened. To appear in Contemp. Math

Scientific paper

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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