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<p>This thesis analyzes the Chow motives of 3 types of smooth projective varieties: the desingularized elliptic self fiber product, the Fano surface of lines on a cubic threefold and an ample hypersurface of an Abelian variety. For the desingularized elliptic self fiber product, we use an isotypic decomposition of the motive to deduce the Murre conjectures. We also prove a result about the intersection product. For the Fano surface of lines, we prove the finite-dimensionality of the Chow motive. Finally, we prove that an ample hypersurface on an Abelian variety possesses a Chow-Kunneth decomposition for which a motivic version of the Lefschetz hyperplane theorem holds.</p> / Dissertation

Wong, Sen. / Thesis submitted in: November 2006. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaf 85). / Abstracts in English and Chinese. / Chapter 1 --- From algebraic sets to Preschemes --- p.5 / Chapter 1.1 --- Affine varieties and prevarieties --- p.5 / Chapter 1.2 --- Affine schemes and preschemes --- p.13 / Chapter 1.3 --- Morphisms --- p.16 / Chapter 1.4 --- Prevarieties vs Preschemes --- p.29 / Chapter 1.5 --- Summary and Products --- p.37 / Chapter 2 --- More on prevarieties --- p.43 / Chapter 2.1 --- Introduction --- p.43 / Chapter 2.2 --- Subprevarieties and product of prevarieties --- p.48 / Chapter 2.3 --- Dominating and birational morphisms --- p.60 / Chapter 2.4 --- Complete varieties --- p.64 / Chapter 2.5 --- Chow's Theorem --- p.68 / Chapter 3 --- From preschemes to schemes --- p.71 / Chapter 3.1 --- Introduction --- p.71 / Chapter 3.2 --- Open subpreschemes and Open immersions --- p.74 / Chapter 3.3 --- Closed subpreschemes and Closed immersions --- p.76 / Chapter 3.4 --- Schemes --- p.79 / Chapter A --- Hilbert's Nullstellensatz --- p.81 / Bibliography --- p.85

The projective plane, PG(2;q), over a Galois field Fq is an incidence structure of points and lines. A (k;n)-arc K in PG(2;q) is a set of k points such that no n+1 of them are collinear but some n are collinear. A (k;n)-arc K in PG(2;q) is called complete if it is not contained in any (k+1;n)-arc. The existence of arcs for particular values of k and n pose interesting problems in finite geometry. It connects with coding theory and graph theory, with important applications in computer science. The main problem, known as the packing problem, is to determine the largest size mn(2;q) of K in PG(2;q). This problem has received much attention. Here, the work establishes complete arcs with a large number of points. In contrast, the problem to determine the smallest size tn(2;q) of a complete (k;n)-arc is mostly based on the lower bound arising from theoretical investigations. This thesis has several goals. The first goal is to classify certain (k;4)-arcs for k = 6,...,38 in PG(2;13). This classification is established through an approach in Chapter 2. This approach uses a new geometrical method; it is a combination of projective inequivalence of (k;4)-arcs up to k = 6 and certain sdinequivalent (k;4)-arcs that have sd-inequivalent classes of secant distributions for k = 7,...,38. The part related to projectively inequivalent (k;4)-arcs up to k=6 starts by fixing the frame points f1;2;3;88g and then classify the projectively inequivalent (5;4)-arcs. Among these (5;4)-arcs and (6;4)-arcs, the lexicographically least set are found. Now, the part regarding sd-inequivalent (k;4)-arcs in this method starts by choosing five sd-inequivalent (7;4)-arcs. This classification method may not produce all sd-inequivalent classes of (k;4)-arcs. However, it was necessary to employ this method due to the increasing number of (k;4)-arcs in PG(2;13) and the extreme computational difficulty of the problem. It reduces the constructed number of (k;4)-arcs in each process for large k. Consequently, it reduces the executed time for the computation which could last for years. Also, this method decreases the memory usage needed for the classification. The largest size of (k;4)-arc established through this method is k = 38. The classification of certain (k;4)-arcs up to projective equivalence, for k = 34,35,36,37,38, is also established. This classification starts from the 77 incomplete (34;4)-arcs that are constructed from the sd-inequivalent (33;4)-arcs given in Section 2.29, Table 2.35. Here, the largest size of (k;4)-arc is still k = 38. In addition, the previous process is re-iterated with a different choice of five sd-inequivalent (7;4)-arcs. The purpose of this choice is to find a new size of complete (k;4)-arc for k > 38. This particular computation of (k;4)-arcs found no complete (k;4)-arc for k > 38. In contrast, a new size of complete (k;4)-arc in PG(2;13) is discovered. This size is k = 36 which is the largest complete (k;4)-arc in this computation. This result raises the second largest size of complete (k;4)-arc found in the first classification from k = 35 to k = 36. The second goal is to discuss the incidence structure of the orbits of the groups of the projectively inequivalent (6;4)-arcs and also the incidence structures of the orbits of the groups other than the identity group of the sd-inequivalent (k;4)-arcs. In Chapter 3, these incidence structures are given for k = 6,7,8,9,10,11,12,13,14,38. Also, the pictures of the geometric configurations of the lines and the points of the orbits are described. The third goal is to find the sizes of certain sd-inequivalent complete (k;4)-arcs in PG(2;13). These sizes of complete (k;4)-arcs are given in Chapter 4 where the smallest size of complete (k;4)-arc is at most k = 24 and the largest size is at least k = 38. The fourth goal is to give an example of an associated non-singular quartic curve C for each complete (k;4)-arc and to discuss the algebraic properties of each curve in terms of the number I of inflexion points, the number jC \K j of rational points on the corresponding arc, and the number N1 of rational points of C . These curves are given in Chapter 5. Also, the algebraic properties of complete arcs of the most interesting sizes investigated in this thesis are studied. In addition, there are two examples of quartic curves C (g0 1) and C (g0 2) attaining the Hasse-Weil- Serre upper bound for the number N1 of rational points on a curve over the finite field of order thirteen. This number is 32. The fifth goal is to classify the (k;4)-arcs in PG(2;13) up to projective inequivalence for k < 10. This classification is established in Chapter 6. It starts by fixing a triad, U1, on the projective line, PG(1;13). Here, the number of projectively inequivalent (k;4)-arcs are tested by using the tool given in Chapter 2. Then, among the number of the projectively inequivalent (10;4)-arcs found, the classification of sd-inequivalent (k;4)-arcs for k = 10 is made. The number of these sd-inequivalent arcs is 36. Then, the 36 sd-inequivalent arcs are extended. The aim here is to investigate if there is a new size of sd-inequivalent (k;4)-arc for k > 38 that can be obtained from these arcs. The largest size of sd-inequivalent (k;4)-arc in this process is the same as the largest size of the sd-inequivalent (k;4)-arc established in Chapter 2, that is, k = 38. In addition, the classification of (k;n)-arcs in PG(2;13) is extended from n = 4 to n = 6. This extension is given in Chapter 7 where some results of the classification of certain (k;6)-arcs for k = 9; : : : ;25 are obtained using the same method as in Chapter 2 for k = 7,...,38. This process starts by fixing a certain (8;6)-arc containing six collinear points in PG(2;13).

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1997. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaf 37). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaf 37). Also available on the Internet.

Hurwitz numbers are a weighted count of degree d ramified covers of curves with specified ramification profiles at marked points on the codomain curve. Isomorphism classes of these covers can be included as a dense open set in a moduli space, called a Hurwitz space. The Hurwitz space has a forgetful morphism to the moduli space of marked, stable curves, and this morphism encodes the Hurwitz numbers. Mikhalkin has constructed a moduli space of tropical marked, stable curves, and this space is a tropical variety. In this paper, I construct a tropical analogue of the Hurwitz space in the sense that it is a connected, polyhedral complex with a morphism to the tropical moduli space of curves such that the degree of the morphism encodes the Hurwitz numbers. / text

In this thesis we apply methods from homological algebra to the study of finite $p$-groups. Let $G$ be a finite $p$-group and let $\mathbb{F}_p$ be the field of $p$ elements. We consider the cohomology groups $\operatorname{H}^1(G,\mathbb{F}_p)$ and $\operatorname{H}^2(G,\mathbb{F}_p)$ and the Massey product structure on these cohomology groups, which we use to deduce properties about $G$. We tie the classical theory of Massey products in with a general method from deformation theory for constructing hulls of functors and see how far the strictly defined Massey products can take us in this setting. We show how these Massey products relate to extensions of modules and to relations, giving us cohomological presentations of $p$-groups. These presentations will be minimal pro-$p$ presentations and will often be different from the presentations we are used to. This enables us to shed some new light on the classification of $p$-groups, in particular we give a `tree construction' illustrating how we can `produce' $p$-groups using cohomological methods. We investigate groups of exponent $p$ and some of the families of groups appearing in the tree. We also investigate the limits of these methods. As an explicit example illustrating the theory we have introduced, we calculate Massey products using the Yoneda cocomplex and give 0-deficiency presentations for split metacyclic $p$-groups using strictly defined Massey products. We also apply these methods to the modular isomorphism problem, i.e. the problem whether (the isomorphism class of) $G$ is determined by $\F_pG$. We give a new class $\mathcal{C}$ of finite $p$-groups which can be distinguished using $\mathbb{F}_pG$.