Irreducible components of quiver Grassmannians

Hubery, Andrew

doi:10.1090/tran/6693

Irreducible components of quiver Grassmannians
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Trans. Amer. Math. Soc. 369 (2017), 1395-1458 Request permission

Abstract:

We consider the action of a smooth, connected group scheme $G$ on a scheme $Y$, and discuss the problem of when the saturation map $\Theta \colon G\times X\to Y$ is separable, where $X\subset Y$ is an irreducible subscheme. We provide sufficient conditions for this in terms of the induced map on the fibres of the conormal bundles to the orbits. Using jet space calculations, one then obtains a criterion for when the scheme-theoretic image of $\Theta$ is an irreducible component of $Y$.

We apply this result to Grassmannians of submodules and several other schemes arising from representations of algebras, thus obtaining a decomposition theorem for their irreducible components in the spirit of the result by Crawley-Boevey and Schröer for module varieties.

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