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Title: Combinatorial Reid's recipe for consistent dimer models
Author: Tapia Amador, Jesus
ISNI:       0000 0004 5368 2412
Awarding Body: University of Bath
Current Institution: University of Bath
Date of Award: 2015
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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 $\Pic(\mathcal{M}_A(\theta))$.
Supervisor: Craw, Alastair Sponsor: Consejo Nacional de Ciencia y Tecnologia
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: dimer models ; Reid's recipe ; toric varieties ; quiver representations