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 $\Pic(\mathcal{M}_A(\theta))$.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:669022 |
| Date | January 2015 |
| Creators | Tapia Amador, Jesus |
| Contributors | Craw, Alastair |
| Publisher | University of Bath |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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