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Developed over more than a century, and still an active area of research today, the classification of algebraic surfaces is an intricate and fascinating branch of mathematics. In this book Professor Beauville gives a lucid and concise account of the subject, following the strategy of F. Enriques, but expressed simply in the language of modern topology and sheaf theory, so as to be accessible to any budding geometer. This volume is self contained and the exercises succeed both in giving the flavour of the extraordinary wealth of examples in the classical subject, and in equipping the reader with most of the techniques needed for research.
Surfaces, Algebraic. --- Algebraic surfaces --- Geometry, Algebraic
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"This book presents methods and results from the theory of Zariski structures and discusses their applications in geometry as well as various other mathematical fields. Its logical approach helps us understand why algebraic geometry is so fundamental throughout mathematics and why the extension to noncommutative geometry, which has been forced by recent developments in quantum physics, is both natural and necessary. Beginning with a crash course in model theory, this book will suit not only model theorists but also readers with a more classical geometric background"--Provided by publisher.
Zariski surfaces. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Surfaces, Zariski --- Surfaces, Algebraic --- Zariski, Oscar,
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We are defining and studying an algebra-geometric analogue of Donaldson invariants by using moduli spaces of semistable sheaves with arbitrary ranks on a polarized projective surface. We are interested in relations among the invariants, which are natural generalizations of the "wall-crossing formula" and the "Witten conjecture" for classical Donaldson invariants. Our goal is to obtain a weaker version of these relations, by systematically using the intrinsic smoothness of moduli spaces. According to the recent excellent work of L. Goettsche, H. Nakajima and K. Yoshioka, the wall-crossing formula for Donaldson invariants of projective surfaces can be deduced from such a weaker result in the rank two case!
Surfaces, Algebraic --- Invariants --- Moduli theory --- Geometry --- Mathematics --- Physical Sciences & Mathematics --- Surfaces, Algebraic. --- Invariants. --- Moduli theory. --- Theory of moduli --- Algebraic surfaces --- Mathematics. --- Algebraic geometry. --- Algebraic Geometry. --- Algebraic geometry --- Math --- Science --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Geometry, algebraic.
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Now back in print, this highly regarded book has been updated to reflect recent advances in the theory of semistable coherent sheaves and their moduli spaces, which include moduli spaces in positive characteristic, moduli spaces of principal bundles and of complexes, Hilbert schemes of points on surfaces, derived categories of coherent sheaves, and moduli spaces of sheaves on Calabi-Yau threefolds. The authors review changes in the field since the publication of the original edition in 1997 and point the reader towards further literature. References have been brought up to date and errors removed. Developed from the authors' lectures, this book is ideal as a text for graduate students as well as a valuable resource for any mathematician with a background in algebraic geometry who wants to learn more about Grothendieck's approach.
Moduli theory --- Sheaf theory --- Surfaces, Algebraic --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Algebraic surfaces --- Cohomology, Sheaf --- Sheaf cohomology --- Sheaves, Theory of --- Sheaves (Algebraic topology) --- Algebraic topology --- Sheaf theory. --- Moduli theory. --- Surfaces, Algebraic.
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Lectures: A. Beauville: Surfaces algébriques complexes.- F.A. Bogomolov: The theory of invariants and its applications to some problems in the algebraic geometry.- E. Bombieri: Methods of algebraic geometry in Char. P and their applications.- Seminars: F. Catanese: Pluricanonical mappings of surfaces with K² =1,2, q=pg=0.- F. Catanese: On a class of surfaces of general type.- I. Dolgacev: Algebraic surfaces with p=pg =0.- A. Tognoli: Some remarks about the "Nullstellensatz".
Surfaces, Algebraic --- Geometry, Algebraic --- Algebraic surfaces --- Mathematics. --- Algebraic geometry. --- Algebraic Geometry. --- Algebraic geometry --- Geometry --- Math --- Science --- Geometry, algebraic.
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The present publication contains a special collection of research and review articles on deformations of surface singularities, that put together serve as an introductory survey of results and methods of the theory, as well as open problems, important examples and connections to other areas of mathematics. The aim is to collect material that will help mathematicians already working or wishing to work in this area to deepen their insight and eliminate the technical barriers in this learning process. This also is supported by review articles providing some global picture and an abundance of examples. Additionally, we introduce some material which emphasizes the newly found relationship with the theory of Stein fillings and symplectic geometry. This links two main theories of mathematics: low dimensional topology and algebraic geometry. The theory of normal surface singularities is a distinguished part of analytic or algebraic geometry with several important results, its own technical machinery, and several open problems. Recently several connections were established with low dimensional topology, symplectic geometry and theory of Stein fillings. This created an intense mathematical activity with spectacular bridges between the two areas. The theory of deformation of singularities is the key object in these connections. .
Curves. --- Singularities (Mathematics). --- Surfaces, Algebraic. --- Deformations of singularities. --- Singularities (Mathematics) --- Research. --- Mathematics. --- Algebraic geometry. --- Algebraic topology. --- Algebraic Topology. --- Algebraic Geometry. --- Geometry, Algebraic --- Geometry, algebraic. --- Algebraic geometry --- Geometry --- Topology
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Resolution of singularities of embedded algebraic surfaces
Algebra. --- Singularities (Mathematics). --- Surfaces, Algebraic. --- 536 --- 536 Heat. Thermodynamics --- Heat. Thermodynamics --- Singularities (Mathematics) --- Geometry, Algebraic --- Algebraic surfaces --- Low temperatures. --- Low temperature research.
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One of the main achievements of algebraic geometry over the last 30 years is the work of Mori and others extending minimal models and the Enriques-Kodaira classification to 3-folds. This book, first published in 2000, is an integrated suite of papers centred around applications of Mori theory to birational geometry. Four of the papers (those by Pukhlikov, Fletcher, Corti, and the long joint paper Corti, Pukhlikov and Reid) work out in detail the theory of birational rigidity of Fano 3-folds; these contributions work for the first time with a representative class of Fano varieties, 3-fold hypersurfaces in weighted projective space, and include an attractive introductory treatment and a wealth of detailed computation of special cases.
Threefolds (Algebraic geometry) --- Geometry, Algebraic. --- Surfaces, Algebraic. --- Algebraic surfaces --- Geometry, Algebraic --- Algebraic geometry --- Geometry --- 3-folds (Algebraic geometry) --- Three-folds (Algebraic geometry) --- Algebraic varieties
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This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry. It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the study of lines and conics. From the modern standpoint, arithmetic is the study of rational and integral points on algebraic varieties over nonclosed fields. A major insight of the 20th century was that arithmetic properties of an algebraic variety are tightly linked to the geometry of rational curves on the variety and how they vary in families. This collection of solicited survey and research papers is intended to serve as an introduction for graduate students and researchers interested in entering the field, and as a source of reference for experts working on related problems. Topics that will be addressed include: birational properties such as rationality, unirationality, and rational connectedness, existence of rational curves in prescribed homology classes, cones of rational curves on rationally connected and Calabi-Yau varieties, as well as related questions within the framework of the Minimal Model Program.
Geometry, Algebraic --- Number theory --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Geometry, Algebraic. --- Surfaces, Algebraic. --- Algebraic surfaces --- Algebraic geometry --- Mathematics. --- Algebraic geometry. --- Geometry. --- Number theory. --- Algebraic Geometry. --- Number Theory. --- Geometry, algebraic. --- Number study --- Numbers, Theory of --- Algebra --- Euclid's Elements
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The problem of enumerating maps (a map is a set of polygonal "countries" on a world of a certain topology, not necessarily the plane or the sphere) is an important problem in mathematics and physics, and it has many applications ranging from statistical physics, geometry, particle physics, telecommunications, biology, ... etc. This problem has been studied by many communities of researchers, mostly combinatorists, probabilists, and physicists. Since 1978, physicists have invented a method called "matrix models" to address that problem, and many results have been obtained. Besides, another important problem in mathematics and physics (in particular string theory), is to count Riemann surfaces. Riemann surfaces of a given topology are parametrized by a finite number of real parameters (called moduli), and the moduli space is a finite dimensional compact manifold or orbifold of complicated topology. The number of Riemann surfaces is the volume of that moduli space. More generally, an important problem in algebraic geometry is to characterize the moduli spaces, by computing not only their volumes, but also other characteristic numbers called intersection numbers. Witten's conjecture (which was first proved by Kontsevich), was the assertion that Riemann surfaces can be obtained as limits of polygonal surfaces (maps), made of a very large number of very small polygons. In other words, the number of maps in a certain limit, should give the intersection numbers of moduli spaces. In this book, we show how that limit takes place. The goal of this book is to explain the "matrix model" method, to show the main results obtained with it, and to compare it with methods used in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions). The book intends to be self-contained and accessible to graduate students, and provides comprehensive proofs, several examples, and gives the general formula for the enumeration of maps on surfaces of any topology. In the end, the link with more general topics such as algebraic geometry, string theory, is discussed, and in particular a proof of the Witten-Kontsevich conjecture is provided.
Geometry, algebraic. --- Geometry --- Mathematics --- Physical Sciences & Mathematics --- Surfaces, Algebraic. --- Riemann surfaces. --- Surfaces, Riemann --- Algebraic surfaces --- Geometry, Algebraic --- Functions --- Combinatorics. --- Algebraic Geometry. --- Combinatorics --- Algebra --- Mathematical analysis --- Algebraic geometry --- Algebraic geometry.
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