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This book consists of two parts. The first is devoted to an introduction to basic concepts in algebraic geometry: affine and projective varieties, some of their main attributes and examples. The second part is devoted to the theory of curves: local properties, affine and projective plane curves, resolution of singularities, linear equivalence of divisors and linear series, Riemann–Roch and Riemann–Hurwitz Theorems. The approach in this book is purely algebraic. The main tool is commutative algebra, from which the needed results are recalled, in most cases with proofs. The prerequisites consist of the knowledge of basics in affine and projective geometry, basic algebraic concepts regarding rings, modules, fields, linear algebra, basic notions in the theory of categories, and some elementary point–set topology. This book can be used as a textbook for an undergraduate course in algebraic geometry. The users of the book are not necessarily intended to become algebraic geometers but may be interested students or researchers who want to have a first smattering in the topic. The book contains several exercises, in which there are more examples and parts of the theory that are not fully developed in the text. Of some exercises, there are solutions at the end of each chapter.

Geometry. --- Projective geometry. --- Algebra. --- Projective Geometry. --- Mathematics --- Mathematical analysis --- Projective geometry --- Geometry, Modern --- Euclid's Elements --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Geometria algebraica --- Geometria algèbrica --- Geometria --- Anàlisi p-àdica --- Cicles algebraics --- Espais algebraics --- Esquemes (Geometria algebraica) --- Esquemes de grups (Matemàtica) --- Geometria algebraica aritmètica --- Grups algebraics lineals --- Geometria analítica --- Geometria biracional --- Geometria enumerativa --- Geometria tropical --- Homologia --- Singularitats (Matemàtica) --- Superfícies algebraiques --- Teoria de mòduls --- Teoria de la intersecció --- Teoria de Hodge --- Varietats abelianes --- Varietats algebraiques --- Corbes algebraiques --- Funcions abelianes

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Algebraic geometry --- 51 --- Mathematics --- 51 Mathematics

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In this volume one finds basic techniques from algebra and number theory (e.g. congruences, unique factorization domains, finite fields, quadratic residues, primality tests, continued fractions, etc.) which in recent years have proven to be extremely useful for applications to cryptography and coding theory. Both cryptography and codes have crucial applications in our daily lives, and they are described here, while the complexity problems that arise in implementing the related numerical algorithms are also taken into due account. Cryptography has been developed in great detail, both in its classical and more recent aspects. In particular public key cryptography is extensively discussed, the use of algebraic geometry, specifically of elliptic curves over finite fields, is illustrated, and a final chapter is devoted to quantum cryptography, which is the new frontier of the field. Coding theory is not discussed in full; however a chapter, sufficient for a good introduction to the subject, has been devoted to linear codes. Each chapter ends with several complements and with an extensive list of exercises, the solutions to most of which are included in the last chapter. Though the book contains advanced material, such as cryptography on elliptic curves, Goppa codes using algebraic curves over finite fields, and the recent AKS polynomial primality test, the authors' objective has been to keep the exposition as self-contained and elementary as possible. Therefore the book will be useful to students and researchers, both in theoretical (e.g. mathematicians) and in applied sciences (e.g. physicists, engineers, computer scientists, etc.) seeking a friendly introduction to the important subjects treated here. The book will also be useful for teachers who intend to give courses on these topics.

Cryptography. --- Electronic books. -- local. --- Number theory. --- Number theory --- Cryptography --- Coding theory --- Ciphers --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Cryptanalysis --- Cryptology --- Secret writing --- Steganography --- Number study --- Numbers, Theory of --- Mathematics. --- Data structures (Computer science). --- Algebra. --- Geometry. --- Combinatorics. --- Number Theory. --- Data Structures, Cryptology and Information Theory. --- Combinatorics --- Mathematical analysis --- Euclid's Elements --- Information structures (Computer science) --- Structures, Data (Computer science) --- Structures, Information (Computer science) --- Electronic data processing --- File organization (Computer science) --- Abstract data types (Computer science) --- Math --- Science --- Signs and symbols --- Symbolism --- Writing --- Data encryption (Computer science) --- Data structures (Computer scienc. --- Data Structures and Information Theory.

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Providing a timely description of the present state of the art of moduli spaces of curves and their geometry, this volume is written in a way which will make it extremely useful both for young people who want to approach this important field, and also for established researchers, who will find references, problems, original expositions, new viewpoints, etc. The book collects the lecture notes of a number of leading algebraic geometers and in particular specialists in the field of moduli spaces of curves and their geometry. This is an important subject in algebraic geometry and complex analysis which has seen spectacular developments in recent decades, with important applications to other parts of mathematics such as birational geometry and enumerative geometry, and to other sciences, including physics.  The themes treated are classical but with a constant look to modern developments (see Cascini, Debarre, Farkas, and Sernesi's contributions), and include very new material, such as Bridgeland stability (see Macri's lecture notes) and tropical geometry (see Chan's lecture notes).

Mathematics. --- Algebraic geometry. --- Category theory (Mathematics). --- Homological algebra. --- Projective geometry. --- Algebraic Geometry. --- Category Theory, Homological Algebra. --- Projective Geometry. --- Projective geometry --- Homological algebra --- Category theory (Mathematics) --- Algebraic geometry --- Math --- Geometry, Modern --- Algebra, Abstract --- Homology theory --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Geometry --- Science --- Moduli theory. --- Curves, Algebraic. --- Algebraic curves --- Algebraic varieties --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Geometry, algebraic. --- Algebra. --- Mathematics --- Mathematical analysis