We can view quiver representations of a fixed dimension vector as an algebraic variety over an algebraically closed field $K$. There is an action of the product of general linear groups on each of these varieties where the orbits of the action correspond to isomorphism classes of quiver representation. A $K$-algebra $A$ is said to have the dense orbit property if for each dimension vector, the product of the general linear group acts on each irreducible component of the module variety with a dense orbit. Under certain conditions, a $K$ algebra $A$ is representation finite if and only if it $A$ has the dense orbit property. The implication representation finite implies the dense orbit property is always true. The converse is not true in general, as shown by Chindris, Kinser, and Weyman in \cite{ryan}. Our main theorem of this thesis builds on their work to give a family of representation infinite algebras with the dense orbit property. We also give a conjectured classification of indecomposables with dense orbits. \par In the future, we hope the work presented here can be used to find even more examples of representation infinite algebra with the dense orbit property to then develop deeper theory to classify algebras with the dense orbit property that are representation infinite.

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 $\Pic(\mathcal{M}_A(\theta))$.

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / A representation of a quiver [Gamma] over a commutative ring R assigns an R-module to each vertex and an R-linear map to each arrow. In this dissertation, we consider R = k[t]/(t[superscript]n) and all R-free representations of [Gamma] which assign a free R-module to each vertex. The category, denoted by Rep[superscript]f[subscript] R([Gamma]), containing all such representations is not an abelian category, but rather an exact category. In this dissertation, we firstly study the Hall algebra of the category Rep[superscript]f[subscript] R([Gamma]), denote by [Eta](R[Gamma]), for a loop-free quiver [Gamma]. A geometric realization of the composition subalgebra of [Eta](R[Gamma]) is given under the framework of Lusztig's geometric setting. Moreover, the canonical basis and a monomial basis of this subalgebra are constructed by using perverse sheaves. This generalizes Lusztig's result about the geometric realization of quantum enveloping algebra. As a byproduct, the relation between this subalgebra and quantum generalized Kac-Moody algebras is obtained. If [Gamma] is a Jordan quiver, which is a quiver with one vertex and one loop, each representation in Rep[superscript]f[subscript] R([Gamma]), gives a matrix over R when we fix a basis of the free R-module. An interesting case arises when considering invertible matrices. It then turns out that one is dealing with representations of the group GL[subscript]m(k[t]/(t[superscript]n)). Character sheaf theory is a geometric character theory of algebraic groups. In this dissertation, we secondly construct character sheaves on GL[subscript]m(k[t]/(t[superscript]2)). Then we define an induction functor and restriction functor on these perverse sheaves.

This thesis constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth toric Fano fourfolds. The tilting bundles lead to a large class of explicit Calabi-Yau-5 algebras, obtained as the corresponding rolled-up helix algebra. We provide two different methods to show that a collection of line bundles is full, whilst the strong exceptional condition is checked using the package QuiversToricVarieties for the computer algebra system Macaulay2, written by the author. A database of the full strong exceptional collections can also be found in this package.

Le but de cette thèse est de construire de nouvelles variétés algébriques complexes de Fano et à canonique triviale dans les espaces homogènes et d'analyser leur géométrie. On commence en construisant les variétés spéciales comme lieux de zéros de fibrés homogènes dans les grassmanniennes généralisées. On donne une complète classification en dimension 4. On prouve que les uniques variétés de dimension 4 hyper-Kahleriennes ainsi construites sont les exemples de Beauville-Donagi et Debarre-Voisin. Le même résultat vaut dans les grassmanniennes ordinaires en toute dimension quand le fibré est irréductible. Ensuite on utilise les lieux de dégénérescence orbitaux (ODL), qui généralisent les lieux de dégénérescence classiques, pour construire d'autres variétés. On rappelle les propriétés basiques des ODL, qu'on définit à partir d'une adhérence d'orbite. On construit trois schémas de Hilbert de deux points sur une K3 comme ODL, et beaucoup d'autres exemples de variétés de Calabi-Yau et de Fano. Puis on étudie les adhérences d'orbites dans les représentations de carquois, et on décrit des effondrements de Kempf pour celles de type A_n et D_4; ceci nous permet de construire davantage de variétés spéciales comme ODL. Pour finir, on analyse les grassmanniennes bisymplectiques, qui sont des Fano particulières. Elles admettent l'action d'un tore avec un nombre fini de points fixes. On étudie leurs petites déformations. Ensuite, on étudie la cohomologie (équivariante) des grassmanniennes symplectiques, qui est utile pour mieux comprendre la cohomologie des grassmanniennes bisymplectiques. On analyse en détail un cas explicite en dimension 6. / The aim of this thesis is to construct new interesting complex algebraic Fano varieties and varieties with trivial canonical bundle and to analyze their geometry. In the first part we construct special varieties as zero loci of homogeneous bundles inside generalized Grassmannians. We give a complete classification for varieties of small dimension when the bundle is completely reducible. Thus, we prove that the only fourfolds with trivial canonical bundle so constructed which are hyper-Kahler are the examples of Beauville-Donagi and Debarre-Voisin. The same holds in ordinary Grassmannians when the bundle is irreducible in any dimension. In the second part we use orbital degeneracy loci (ODL), which are a generalization of classical degeneracy loci, to construct new varieties. ODL are constructed from a model, which is usually an orbit closure inside a representation. We recall the fundamental properties of ODL. As an illustration of the construction, we construct three Hilbert schemes of two points on a K3 surface as ODL, and many examples of Calabi-Yau and Fano threefolds and fourfolds. Then we study orbit closures inside quiver representations, and we provide crepant Kempf collapsings for those of type A_n, D_4; this allows us to construct some special varieties as ODL.Finally we focus on a particular class of Fano varieties, namely bisymplectic Grassmannians. These varieties admit the action of a torus with a finite number of fixed points. We find the dimension of their moduli space. We then study the equivariant cohomology of symplectic Grassmannians, which turns out to help understanding better that of bisymplectic ones. We analyze in detail the case of dimension 6.