The aim of this thesis is to generalise Reid's recipe as first defined by Reid for$G-\Hilb(\mathbb{C}^3)$ ($G$ a finite abelian subgroup of $\SL(3, \mathbb{C})$) to the setting of consistent dimer models. We study the $\theta$-stable representations of a quiver $Q$ with relations $\mathcal{R}$ dual to a consistent dimer model $\Gamma$ in order to introduce a well-defined recipe that marks interior lattice points and interior line segments of a cross-section of the toric fan $\Sigma$ of the moduli space $\mathcal{M}_A(\theta)$ with vertices of $Q$, where $A=\mathbb{C}Q/\langle \mathcal{R}\rangle$. After analysing the behaviour of 'meandering walks' on a consistent dimer model $\Gamma$ and assuming two technical conjectures, we introduce an algorithm - the arrow contraction algorithm - that allows us to produce new consistent dimer models from old. This algorithm could be used in the future to show that in doing combinatorial Reid's recipe, every vertex of $Q$ appears 'once' and that combinatorial Reid's recipe encodes the relations of the tautological line bundles of $\mathcal{M}_A(\theta)$ in $\Pic(\mathcal{M}_A(\theta))$.

Date of Award | 30 Jun 2015 |
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Original language | English |
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Awarding Institution | |
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Sponsors | Consejo Nacional de Ciencia y Tecnologia |
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Supervisor | Alastair Craw (Supervisor) |
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- dimer models
- Reid's recipe
- toric varieties
- quiver representations

Combinatorial Reid's Recipe for Consistent Dimer Models

Tapia Amador, J. (Author). 30 Jun 2015

Student thesis: Doctoral Thesis › PhD