We consider smooth complex projective varieties X which are rationally connected by rational curves of degree d with respect to a fixed ample line bundle L on X, and we focus our attention on conic connected manifolds (d=2) and on rationally cubic connected manifolds (d=3). Conic connected manifolds were studied by Ionescu and Russo; they considered conic connected manifolds embedded in projective space (i.e. L is very ample) and they proved a classification theorem for these manifolds. We show that their classification result holds true assuming just the ampleness of L. Moreover we give a different proof of a theorem due to Kachi and Sato; this result characterizes a special subclass of conic connected manifolds. As already said before, we study also rationally cubic connected manifolds. We prove that if rationally cubic connected manifolds are covered by “lines†, i.e. by curves of degree 1 with respect to L, then the Picard number of X is equal to or less than 3; moreover we show that if the Picard number is equal to 3 then there is a covering family of “lines†whose numerical class spans a negative extremal ray of the Kleiman-Mori cone of X. Unfortunately, for rationally cubic connected manifolds which don't admit a covering family of “lines†there isn't an upper bound on the Picard number. However we prove that if we consider rationally cubic connected manifolds which are not covered by “lines†but are Fano then up to a few exceptions in dimension 2 also the Picard number of these manifolds is equal to or less than 3. In particular, supposing that the dimension of X is greater than 2, we show that either the Picard number is equal to or less than 2 or X is the blow up of projective space along two disjoint subvarieties that are linear subspaces or quadrics.

In this dissertation I face three main arguments. 1) Classification of Fano-Mori contractions. 2) Chern numbers on smooth threefolds. 3) Pluricanonical systems.

Galois spaces, that is affine and projective spaces of dimension N ≥ 2 defined over a finite (Galois) field F_q, are well known to be rich of nice geometric, combinatorial and group theoretic properties that have also found wide and relevant applications in several branches of Combinatorics, as well as in more practical areas, notably Coding Theory and Cryptography. The systematic study of Galois spaces was initiated in the late 1950’s by the pioneering work of B. Segre [59]. The trilogy [34, 36, 42] covers the general theory of Galois spaces including the study of objects which are linked to linear codes. Typical such objects are plane arcs and their generalizations - especially caps and arcs in higher dimensions - whose code theoretic counterparts are distinguished types of error-correcting and covering linear codes. Their investigation has received a great stimulus from Coding Theory, especially in the last decades; see the survey papers [40, 41]. An important issue in this context is to ask for explicit constructions of small complete arcs and small complete caps. A cap in a Galois space is a set of points no three of which are collinear. A cap is complete if its secants (lines through two points of the set) cover the whole space. An arc in a Galois space of dimension N is a set of points no N+1 of which lying on the same hyperplane. In analogy with caps, an arc which is maximal with respect to set-theoretical inclusion is said to be complete. Also, arcs coincide with caps in Galois planes. From these geometrical objects, there arise linear codes which turn out to have very good covering properties, provided that the size of the set is small with respect to the dimension N and the order q of the ambient space. For the size t(AG(N,q)) of the smallest complete caps in a Galois affine space AG(N,q) of dimension N over F_q, the trivial lower bound is √2q^{N−1/2}. General constructions of complete caps whose size is close to this lower bound are only known for q even and N odd, see [16, 19, 29, 52]. When N is even, complete caps of size of the same order of magnitude as cq^{N/2}, with c a constant independent of q, are known to exist for both the odd and the even order case, see [16, 18, 28, 29, 31] (see also Section 2.2 and the references therein). Whereas, few constructions of small complete arcs in Galois spaces of dimension N>2 are known. In [65, 66, 67], small complete arcs having many points in common with the normal rational curve are investigated (see Section 4.2.3 for comparisons with our results). In this thesis, new infinite families of complete arcs and caps in higher dimensional spaces are constructed from algebraic curves defined over a finite field. In most cases, no smallest complete caps/arcs were previously known in the literature. Although caps and arcs are rather combinatorial objects, constructions and proofs sometimes heavily rely on concepts and results from Algebraic Geometry in positive characteristic.

Braid theory is a very active research field. Braids can be studied from variuos points of view and have applications in different fields, e.g. in mathematical physics and in biology. In this thesis we provide a formal introduction to some topics in the mathematical theory of braids and two possible approaches to this field at a popular level: a movie and a workshop. The scientific movie addressed to a non-specialist audience has been realized using the free ray-tracer POV-Ray. It is divided into four parts, each of which has a length of about 15 minutes. The content ranges from the introduction of basic concepts to deep results. The workshop activity is based on the action of braids on loops and aims to invite and lead the audience to a mathematical formalization of the principal concepts involved: braids, curves and group actions.

We consider edge-colorings and flows problems in Graph Theory that are hard to solve for Class 2 graphs. Most of them are strongly related to some outstanding open conjectures, such as the Cycle Double Cover Conjecture, the Berge-Fulkerson Conjecture, the Petersen Coloring Conjecture and the Tutte's 5-flow Conjecture. We obtain some new restrictions on the structure of a possible minimum counterexample to the former two conjectures. We prove that the Petersen graph is, in a specific sense, the only graph that could appear in the Petersen Coloring Conjecture, and we provide evidence that led to propose an analogous of the Tutte's 5-flow conjecture in higher dimensions. We prove a characterization result and a sufficient condition for general graphs in relation to another edge-coloring problem, which is the determination of the palette index of a graph.

Given a birational map among projective varieties, it is known that there exists a variety Z with a one-dimensional torus action such that the birational map is induced from two geometric quotients of Z. We proceed in the opposite direction: given a smooth projective variety X with a one-dimensional torus action, one can define a birational map associated to the action and study the properties of the map via the geometry of X. Rational homogeneous varieties admit natural torus actions, so they are a good class of example to test the general theory. In the thesis, we obtain and discuss some results about the birational maps associated to some one-dimensional torus actions on rational homogeneous varieties.

In the first part of this thesis we report results about some “linear” trapdoors that can be embedded in a block cipher. In particular we are interested in any block cipher which has invertible S-boxes and that acts as a permutation on the message space, once the key is chosen. The message space is a vector space and we can endow it with alternative operations (hidden sums) for which the structure of vector space is preserved. Each of this operation is related to a different copy of the affine group. So, our block cipher could be affine with respect to one of these hidden sums. We show conditions on the S-box able to prevent a type of trapdoors based on hidden sums, in particular we introduce the notion of Anti-Crooked function. Moreover we shows some properties of the translation groups related to these hidden sums, characterizing those that are generated by affine permutations. In that case we prove that hidden sum trapdoors are practical and we can perform a global reconstruction attack. We also analyze the role of the mixing layer obtaining results suggesting the possibility to have undetectable hidden sum trapdoors using MDS mixing layers. In the second part we take into account the index coding with side information (ICSI) problem. Firstly we investigate the optimal length of a linear index code, that is equal to the min-rank of the hypergraph related to the instance of the ICSI problem. In particular we extend the the so-called Sandwich Property from graphs to hypergraphs and also we give an upper bound on the min-rank of an hypergraph taking advantage of incidence structures such as 2-designs and projective planes. Then we consider the more general case when the side information are coded, the index coding with coded side information (ICCSI) problem. We extend some results on the error correction index codes to the ICCSI problem case and a syndrome decoding algorithm is also given.

This thesis deals with the classical problem of prime numbers represented by polynomials. It consists of three parts. In the first part I collected many results about the problem. Some of them are quite recent and this part can be considered as a survey of the state of the art of the subject. In the second part I present two results due to P. Pleasants about the cubic polynomials with integer coefficients in several variables. The aim of this part is to simplify the works of Pleasants and modernize the notation employed. In such a way these important theorems are now in a more readable form. In the third part I present some original results related with some algebraic invariants which are the key-tools in the works of Pleasants. The hidden diophantine nature of these invariants makes them very difficult to study. Anyway some results are proved. These results make the results of Pleasants somewhat more effective.

In this thesis, I show that the representation of prime integers by reduced binary quadratic forms of given discriminant can be obtained in polynomial complexity using Schoof's algorithm for counting the number of points of elliptic curves over finite fields. It is a remarkable fact that, although an algorithm of Gauss' solved the representation problem long time ago, a solution in polynomial complexity is very recent and almost unnoticed in the literature. Further, I present a viable alternative to Gauss' algorithm, which constitutes the main original contribution of my thesis. This alternative way of computing in polynomial time an explicit solution of the representation problem is particularly suitable whenever the number of not equivalent reduced forms is small. Lastly, I report that, in the efforts of improving Schoof's algorithm, a marginal incompleteness in its original formulation was identified. This weakness was eliminated by a slight modification of the algorithm suggested by Schoof himself.

In this thesis, we study the degree of the canonical map of surfaces of general type. In particular, we give the first examples known in the literature of surfaces having degree d=10,11, 13, 14, 15, and 18 of the canonical map. They are presented in a self-contained and independent way from the rest of the thesis. We show also how we have discovered them. These surfaces are product-quotient surfaces. In this thesis, we study the theory of product-quotient surfaces giving also some new results and improvements. As a consequence of this, we have written and run a MAGMA script to produce a list of families of product-quotient surfaces having geometric genus three and a self-intersection of the canonical divisor large. After that, we study the canonical map of product-quotient surfaces and we apply the obtained results to the list of product-quotient surfaces just mentioned. In this way, we have discovered the examples of surfaces having degree d=10,11,14, and 18 of the canonical map. The remaining ones with degrees 13 and 15 do not satisfy the assumptions to compute the degree of the canonical map directly. Hence we have had to compute the canonical degree of these two families of product-quotient surfaces in a very explicit way through the equations of the pair of curves defining them. Another work of this thesis is the classification of all smooth surfaces of general type with geometric genus three which admits an action of a group G isomorphic to \mathbb Z_2^k and such that the quotient is a projective plane. This classification is attained through the theory of abelian covers. We obtained in total eleven families of surfaces. We compute the canonical map of all of them, finding in particular a family of surfaces with a canonical map of degree 16 not in the literature. We discuss the quotients by all subgroups of G finding several K3 surfaces with symplectic involutions. In particular, we show that six families are families of triple K3 burgers in the sense of Laterveer. Finally, in another work we study also the possible accumulation points for the slopes K^2/ \chi of unbounded sequences of minimal surfaces of general type having a degree d of the canonical map. As a new result, we construct unbounded families of minimal (product-quotient) surfaces of general type whose degree of the canonical map is 4 and such that the limits of the slopes K^2/ \chi assume countably many different values in the closed interval [6+2/3, 8].