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On the equations and classification of toric quiver varieties

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

Published online by Cambridge University Press:  03 March 2016

Mátyás Domokos  and
Dániel Joó
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

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.

Keywords

binomial idealquiver representationmoduli spaceflow polytope

MSC classification

Primary: 14M25: Toric varieties, Newton polyhedra
Secondary: 14L24: Geometric invariant theory 16G20: Representations of quivers and partially ordered sets 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 2 , April 2016 , pp. 265 - 295
Copyright
Copyright © Royal Society of Edinburgh 2016 

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