In this thesis, we first summarize the known results about lifting algebraic surfaces in characteristic p &gt / 0 to characteristic zero, and then we study lifting fibrations on these surfaces to characteristic zero. We prove that fibrations on ruled surfaces, the natural fibration on Enriques surfaces of classical type, the induced fibration on K3-surfaces covering these types of Enriques surfaces, and fibrations on certain hyperelliptic and quasi-hyperelliptic surfaces lift. We also obtain some fragmentary results concerning the smooth isotrivial fibrations and the fibrations on surfaces of Kodaira dimension 1.

In this thesis, we study the problem of lifting fibrations on surfaces in characteristic p, to characteristic zero. We restrict ourselves mainly to the case of natural fibrations on surfaces with Kodaira dimension -1 or 0. We determine whether such a fibration lifts to characteristic zero. Then, we try to find the smallest ring over which a lifting is possible. Finally,in some favourable cases, we compare the moduli of liftings of the fibration to the moduli of liftings of the surface under consideration.

In the first three chapters of this thesis we study two conjectures relating arithmetic with geometry, namely Tate and Lang&rsquo / s conjectures, for a certain class of algebraic surfaces. The surfaces we are interested in are assumed to be defined over a number field, have irregularity two and admit a genus two fibration over an elliptic curve. In the final chapter of the thesis we prove the isomorphism of the Picard motives of an arbitrary variety and its Albanese variety.

In this thesis, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that, a surface admitting a smooth fibration as above is elliptic and we employ results on the moduli of polarized elliptic surfaces, to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes H(1,X(d),n) of morphisms of degree n from elliptic curves to the modular curve X(d), d&amp / #8804 / 3. Ultimately, we show that the moduli spaces, considered, are fiber spaces over the affine line A&sup1 / with fibers determined by the components of H (1,X(d),n).

A new normal form x2 + y2 = c2(1 + x2y2) of elliptic curves was introduced by M. Harold Edwards in 2007 over the field k having characteristic different than 2. This new form has very special and important properties such that addition operation is strongly unified and complete for properly chosen parameter c . In other words, doubling can be done by using the addition formula and any two points on the curve can be added by the addition formula without exception. D. Bernstein and T. Lange added one more parameter d to the normal form to cover a large class of elliptic curves, x2 + y2 = c2(1 + dx2y2) over the same field. In this thesis, an expository overview of the literature on Edwards curves is given. First, the types of Edwards curves over the nonbinary field k are introduced, addition and doubling over the curves are derived and efficient algorithms for addition and doubling are stated with their costs. Finally, known elliptic curves and Edwards curves are compared according to their cryptographic applications. The way to choose the Edwards curve which is most appropriate for cryptographic applications is also explained.

A net is a special configuration of lines and points in the projective plane. There are certain restrictions on the number of its lines and points. We proved that there cannot be any (4,4) nets in CP^2. In order to show this, we use tropical algebraic geometry. We tropicalize the hypothetical net and show that there cannot be such a configuration in CP^2.

The aim of this thesis is to study the Hilbert function of a one-dimensional Gorenstein local ring of embedding dimension four in the case of monomial curves. We show that the Hilbert function is non-decreasing for some families of Gorenstein monomial curves in affine 4-space. In order to prove this result, under some arithmetic assumptions on generators of the defining ideal, we determine the minimal generators of their tangent cones by using the standard basis and check the Cohen-Macaulayness of them. Later, we determine the behavior of the Hilbert function of these curves, and we extend these families to higher dimensions by using a method developed by Morales. In this way, we obtain large families of local rings with non-decreasing Hilbert function.

abstract HOMOLOGY OF REAL ALGEBRAIC VARIETIES AND MORPHISMS TO SPHERES &uml / OZT&uml / URK, Ali Ph.D., Department of Mathematics Supervisor: Assoc. Prof. Dr. Yildiray OZAN August 2005, 24 pages Let X and Y be affine nonsingular real algebraic varieties. One of the classical problems in real algebraic geometry is whether a given C1 mapping f : X ! Y can be approximated by regular mappings in the space of C1 mappings. In this thesis, we obtain some sufficient conditions in the case when Y is the standard sphere Sn. In the second part of the thesis, we study mainly the kernel of the induced map on homology i : Hk(X,R) ! Hk(XC,R), where i : X ! XC is a nonsingular projective complexification. First, using Lefshcetz Hyperplane Section Theorem we study KHk(X H,R), where H is a hyperplane. In the remaining part, we relate KHk(X,R) to the realization of cohomology classes of XC by harmonic forms.

The aim of this work is to study the interaction between two classical objects of mathematics: the modular group, and the absolute Galois group. The latter acts on the category of finite index subgroups of the modular group. However, it is a task out of reach do understand this action in this generality. We propose a lattice which parametrizes a certain system of &rdquo / geometric&rdquo / elements in this category. This system is setwise invariant under the Galois action, and there is a hope that one can explicitly understand the pointwise action on the elements of this system. These elements admit moreover a combinatorial description as quadrangulations of the sphere, satisfying a natural nonnegative curvature condition. Furthermore, their connections with hypergeometric functions allow us to realize these quadrangulations as points in the moduli space of rational curves with 8 punctures. These points are conjecturally defined over a number field and our ultimate wish is to compare the Galois action on the lattice elements in the category and the corresponding points in the moduli space.