On the equations and classification of toric quiver varieties

Mátyás Domokos; Dániel Joó

doi:10.1017/S0308210515000529

On the equations and classification of toric quiver varieties

Part of: Algebraic groups Special varieties Representation theory of rings and algebras Polytopes and polyhedra

Published online by Cambridge University Press: 03 March 2016

Mátyás Domokos and

Dániel Joó

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Mátyás Domokos: Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13–15, 1053 Budapest, Hungary and Department of Mathematics, Central European University, Nádor utca 9, 1051 Budapest, Hungary (domokos.matyas@renyi.mta.hu; joo.daniel@renyi.mta.hu)
Dániel Joó: Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13–15, 1053 Budapest, Hungary and Department of Mathematics, Central European University, Nádor utca 9, 1051 Budapest, Hungary (domokos.matyas@renyi.mta.hu; joo.daniel@renyi.mta.hu)

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Abstract

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Abstract

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Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight, there is an associated quasi-projective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3. In each fixed dimension d up to isomorphism there are only finitely many d-dimensional toric quiver varieties. A procedure for their classification is outlined.