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The basic problem of deformation theory in algebraic geometry involves watching a small deformation of one member of a family of objects, such as varieties, or subschemes in a fixed space, or vector bundles on a fixed scheme. In this new book, Robin Hartshorne studies first what happens over small infinitesimal deformations, and then gradually builds up to more global situations, using methods pioneered by Kodaira and Spencer in the complex analytic case, and adapted and expanded in algebraic geometry by Grothendieck. Topics include: * deformations over the dual numbers; * smoothness and the infinitesimal lifting property; * Zariski tangent space and obstructions to deformation problems; * pro-representable functors of Schlessinger; * infinitesimal study of moduli spaces such as the Hilbert scheme, Picard scheme, moduli of curves, and moduli of stable vector bundles. The author includes numerous exercises, as well as important examples illustrating various aspects of the theory. This text is based on a graduate course taught by the author at the University of California, Berkeley.
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This book focuses on recent advances in the classification of complex projective varieties. It is divided into two parts. The first part gives a detailed account of recent results in the minimal model program. In particular, it contains a complete proof of the theorems on the existence of flips, on the existence of minimal models for varieties of log general type and of the finite generation of the canonical ring. The second part is an introduction to the theory of moduli spaces. It includes topics such as representing and moduli functors, Hilbert schemes, the boundedness, local closedness and separatedness of moduli spaces and the boundedness for varieties of general type. The book is aimed at advanced graduate students and researchers in algebraic geometry.
Geometry --- landmeetkunde --- Algebraic geometry. --- Algebraic Geometry. --- Algebraic geometry
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This is a reference book for academics working in the fields of several complex variables, PDE theory and geometry. The book collects articles on most recent developments in these fields with special emphasis on interactions between them. It includes introductory and expository presentations of the relevant tools, techniques and objects for graduate and PhD students with a corresponding specialization. The volume comprises contributions of leading experts in the field written in a comprehensive and accessible style. Contributors: D. Barlet, S. Berhanu, A. Bove, D.W. Catlin, J.P. D'Angelo, M. Derridj, F. Forstneric, S. Fu, K. Gansberger, P. Guan, F. Haslinger, B. Helffer, J. Hounie, N. Hungerbühler, F. Lárusson, Ch. Laurent-Thiébaut, L. Lempert, H.-M. Maire, G.A. Mendoza, A. Meziani, N. Mok, M. Mughetti L. Ni, M.-C. Shaw, D.S. Tartakoff, D. Zaitsev
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This is a reference book for academics working in the fields of several complex variables, PDE theory and geometry. The book collects articles on most recent developments in these fields with special emphasis on interactions between them. It includes introductory and expository presentations of the relevant tools, techniques and objects for graduate and PhD students with a corresponding specialization. The volume comprises contributions of leading experts in the field written in a comprehensive and accessible style. Contributors: D. Barlet, S. Berhanu, A. Bove, D.W. Catlin, J.P. D’Angelo, M. Derridj, F. Forstnerič, S. Fu, K. Gansberger, P. Guan, F. Haslinger, B. Helffer, J. Hounie, N. Hungerbühler, F. Lárusson, Ch. Laurent-Thiébaut, L. Lempert, H.-M. Maire, G.A. Mendoza, A. Meziani, N. Mok, M. Mughetti L. Ni, M.-C. Shaw, D.S. Tartakoff, D. Zaitsev.
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This book introduces the reader to modern algebraic geometry. It presents Grothendieck's technically demanding language of schemes that is the basis of the most important developments in the last fifty years within this area. A systematic treatment and motivation of the theory is emphasized, using concrete examples to illustrate its usefulness. Several examples from the realm of Hilbert modular surfaces and of determinantal varieties are used methodically to discuss the covered techniques. Thus the reader experiences that the further development of the theory yields an ever better understanding of these fascinating objects. The text is complemented by many exercises that serve to check the comprehension of the text, treat further examples, or give an outlook on further results. The volume at hand is an introduction to schemes. To get startet, it requires only basic knowledge in abstract algebra and topology. Essential facts from commutative algebra are assembled in an appendix. It will be complemented by a second volume on the cohomology of schemes. Prevarieties - Spectrum of a Ring - Schemes - Fiber products - Schemes over fields - Local properties of schemes - Quasi-coherent modules - Representable functors - Separated morphisms - Finiteness Conditions - Vector bundles - Affine and proper morphisms - Projective morphisms - Flat morphisms and dimension - One-dimensional schemes - Examples Prof. Dr. Ulrich Görtz, Institute of Experimental Mathematics, University Duisburg-Essen Prof. Dr. Torsten Wedhorn, Department of Mathematics, University of Paderborn.
Mathematics. --- Algebra. --- Mathématiques --- Algèbre --- Géométrie algébrique --- Geometric geometry. --- Geometry, algebraic. --- Algebraic Geometry. --- Mathematics --- Mathematical analysis --- Algebraic geometry --- Geometry --- Geometry, Algebraic. --- Schemes (Algebraic geometry) --- Geometry, Algebraic --- Algebraic geometry.
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Federico Gaeta (1923–2007) was a Spanish algebraic geometer who was a student of Severi. He is considered to be one of the founders of linkage theory, on which he published several key papers. After many years abroad he came back to Spain in the 1980s. He spent his last period as a professor at Universidad Complutense de Madrid. In gratitude to him, some of his personal and mathematically close persons during this last station, all of whom bene?ted in one way or another by his ins- ration, have joined to edit this volume to keep his memory alive. We o?er in it surveys and original articles on the three main subjects of Gaeta’s interest through his mathematical life. The volume opens with a personal semblance by Ignacio Sols and a historical presentation by Ciro Ciliberto of Gaeta’s Italian period. Then it is divided into three parts, each of them devoted to a speci?c subject studied by Gaeta and coordinated by one of the editors. For each part, we had the advice of another colleague of Federico linked to that particular subject, who also contributed with a short survey. The ?rst part, coordinated by E. Arrondo with the advice of R.M.
Geometry, Algebraic. --- Invariants. --- Liaison theory (Mathematics). --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Algebraic geometry --- Mathematics. --- Algebraic geometry. --- Algebraic Geometry. --- Geometry, algebraic.
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