Thesis (Ph.D.)--University of Nebraska-Lincoln, 2009. / Title from title screen (site viewed January 5, 2010). PDF text: vi, 65 p. : ill. ; 645 K. UMI publication number: AAT 3359857. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (Ph. D.)--Northeastern University, 2008. / Title from title page (viewed March 26, 2009). Graduate School of Arts and Sciences, Dept. of Mathematics. Includes bibliographical references (p.80-82).

Thesis (Ph. D.)--Missouri University of Science and Technology, 2009. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed April 29, 2009) Includes bibliographical references (p. 142-144).

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2009. / Title from title screen (site viewed January 5, 2010). PDF text: v, 69 p. ; 721 K. UMI publication number: AAT 3360497. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 80-84). Available also in a digital version from Dissertation Abstracts.

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

L'objectif de cette thèse est d'étudier les propriétés de variétés toriques particulières. La thèse est divisée en trois parties, les deux premières étant fortement liées. Dans la première partie, nous étudions des variétés algébriques associées aux processus de Markov sur les arbres. A chaque processus de Markov sur un arbre on peut associer une variété algébrique. Motivé par la biologie, nous nous concentrons sur les processus de Markov dé finis par une action de groupe. Nous étudions les conditions pour que la variété obtenue soit torique. Nous donnons un résultat où les variétés obtenues sont normales, ainsi que des exemples où elles ne le sont pas. L'une des principales méthodes que nous utilisons est la généralisation des notions de prises et de réseaux introduites dans [BW07] à des groupes abéliens arbitraires. Dans notre contexte, les réseaux forment un groupe qui agit sur la variété. Par ailleurs, l'espace ambiant de lavariété est la représentation régulière de ce groupe. Le principal problème ouvert que nous essayons de résoudre dans cette partie est une conjecture de Sturmfels et Sullivant [SS05, Conjecture 2] indiquant que le schéma a fine associé au modèle 3-Kimura estdé fini par un idéal engendré en degré 4. Notre meilleur résultat dit que le schéma projectif associé peut être dé fini par un idéal engendré en degré 4. Avec Maria Donten -Bury, nous proposons une méthode pour engendrer l'idéal associé à la variété pour tous les modèles. Nous montrons que notre méthode fonctionne pour de nombreux modèles ainsi que pour les arbres si et seulement si la conjecture de Sturmfels et Sullivant est vraie. Nous présentons quelques applications, par exemple au problème d'identi abilité en biologie. La deuxième partie concerne les variétés algébriques associées aux graphes trivalents pour le modèle de Jukes-Cantor binaire. Il s'agit d'un travail en commun avec Weronika Buczyńska, Jarosław Buczyński et Kaie Kubjas. La variété associée á un graphe peut être représentéevpar un semi-groupe gradué. Nous étudions les liens entre les propriétés du graphe et le semigroupe. Le théorème principal borne le degré en lequel le semi-groupe est engendré par le premier nombre de Betti du graphe, plus un. Dans la dernière partie, nous étudions la structure de la catégorie dérivée des faisceaux cohérents des variétés toriques lisses. Dans un travail commun avec Michał Lasoń [LM11], nous construisons une collection fortement exceptionnelle complète de fi brés en droites pour une grande classe de variétés toriques complètes lisses dont le nombre de Picard est égal á trois. De nombreuses questions concernant le type de collections auxquelles on peut s'attendre sur les variétés toriques de certains types sont encore ouvertes. A ce titre, nous prouvons que Pn éclaté en deux points ne possède pas de collection fortement exceptionnelle complète de fibrés en droites pour n assez grand. Ceci fournit une collection infi nie de contre-exemples à la conjecture de King. Le premier contre-exemple est dû à Hille et Perling [HP06]. Récemment, des contre-exemples ont également été trouvés par E mov [E ] dans le cadre des variétés de Fano. Nous allons travailler sur le corps des nombres complexes C. Toutes les variétés considérées sont des variétés algébriques dans le sens de [Har77]. / The aim of this thesis is to investigate the properties of special toric varieties. The thesis is divided into three parts. The first two of them are strongly related to each other.In the fi rst, main part we study algebraic varieties associated to Markov processes on trees. To each Markov process on a tree one can associate an algebraic variety. Motivated by biology, we focus on Markov processes de fined by a group action. We investigate underwhich conditions the obtained variety is toric. We provide conditions ensuring that the obtained varieties are normal, as well as give examples when they are not. One of the main tools we use is the generalization of the notions of sockets and networks introduced in [BW07] to arbitrary abelian groups. In our setting the networks form a group, that acts on the variety. Moreover the ambient space of the variety is the regular representation of this group. The main open problem that we address in this part is a conjecture of Sturmfels and Sullivant [SS05, Conjecture 2] stating that the afi ne scheme associated to the 3-Kimura model is de fined by an ideal generated in degree 4. Our strongest result states that the associated projective scheme can be generated in degree 4. Together with Maria Donten -Bury we also propose a method for generating the ideal defi ning the variety for any model. We prove that our method works for many models and trees if and only if the conjecture of Sturmfels and Sullivant holds. We present some applications, for example to theidenti ability problem in biology. The second part concerns algebraic varieties associated to trivalent graphs for the binary Jukes-Cantor model. It is a joint work with Weronika Buczyńska, Jarosław Buczyński and Kaie Kubjas. In case of the graph, the associated variety can be represented by a graded semigroup. We investigate the connections between properties of the graph and the semigroup. The main theorem bounds the degree in which the semigroup is generated by the first Betti number of the graph plus one. Due to connections with the first part much of the terminology that we use is either a specialization or generalization of previous de finitions. From the one hand, as we are working with graphs with possible loops the notions of leaves, nodes and valency are more subtile than for trees. From the other hand, as we are dealing only with the binary Jukes-Cantor model, sockets and networks have got a very special form. In the last part we study the structure of the derived category of coherent sheaves for smooth toric varieties. As a result of a joint work with Michał Lasoń [LM11] we construct a full, strongly exceptional collection of line bundles for a large class of smooth, complete toric varieties with Picard number three. Many questions concerning what kind of collections should be expected on toric varieties of certain types are still open. As a contribution we prove that Pn blown up in two points does not have a full, strongly exceptional collection of line bundles for n large enough. This provides an in finite collection of counterexamples to King's conjecture. The first such counterexample is due to Hille andPerling [HP06]. Recently also counterexamples in the Fano case were found by E mov [E ].

A polyhedron P is a subset of a rational vector space V bounded by hyperplanes. If we fix a lattice in V , then we may consider the exponential integral and sum, two meromorphic functions on the dual vector space which serve to generalize the notion of volume of and number of lattice points contained in P, respectively. In 2007, Berline and Vergne constructed an Euler-Maclaurin formula that relates the exponential sum of a given polyhedron to the exponential integral of each face. This formula was "local", meaning that the coefficients in this formula had certain properties independent of the given polyhedron. In this dissertation, the author finds a new construction for this formula which is very different from that of Berline and Vergne. We may 'perturb' any polyhedron by tranlsating its bounding hyperplanes. The author defines a ring of differential operators R(P) on the exponential volume of the perturbed polyhedron. This definition is inspired by methods in the theory of toric varieties, although no knowledge of toric varieties is necessary to understand the construction or the resulting Euler-Maclaurin formula. Each polyhedron corresponds to a toric variety, and there is a dictionary between combinatorial properties of the polyhedron and algebro-geometric properties of this variety. In particular, the equivariant cohomology ring and the group of equivariant algebraic cycles on the corresponding toric variety are equal to a quotient ring and subgroup of R(P), respectively. Given an inner product (or, more generally, a complement map) on V , there is a canonical section of the equivariant cohomology ring into the group of algebraic cycles. One can use the image under this section of a particular differential operator called the Todd class to define the Euler-Maclaurin formula. The author shows that this formula satisfies the same properties which characterize the Berline-Vergne formula.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 113-117). Available also in a digital version from Dissertation Abstracts.