Combinatorial Reid's Recipe for Consistent Dimer Models

  • Jesus Tapia Amador

Student thesis: Doctoral ThesisPhD

Abstract

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 Award30 Jun 2015
LanguageEnglish
Awarding Institution
  • University of Bath
SponsorsConsejo Nacional de Ciencia y Tecnologia
SupervisorAlastair Craw (Supervisor)

Keywords

  • dimer models
  • Reid's recipe
  • toric varieties
  • quiver representations

Cite this

Combinatorial Reid's Recipe for Consistent Dimer Models
Tapia Amador, J. (Author). 30 Jun 2015

Student thesis: Doctoral ThesisPhD