Arithmetic Geometry Of Toric Varieties
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Arithmetic Geometry Of Toric Varieties
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Author : José Ignacio Burgos Gil
language : en
Publisher:
Release Date : 2014
Arithmetic Geometry Of Toric Varieties written by José Ignacio Burgos Gil and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with Mathematics categories.
The authors show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, the authors study the Arakelov geometry of toric varieties. In particular, they consider models over a discrete valuation ring, metrized line bundles, and their associated measures and heights. They show that these notions can be translated in terms of convex analysis and are closely related to objects such as polyhedral complexes, concave functions, real Monge-Ampere measures, and Legendre-Fenchel duality. The authors also present a closed formula for the integral over a polytope of a function of one variable composed with a linear form. This formula allows them to compute the height of toric varieties with respect to some interesting metrics arising from polytopes and compute the height of toric projective curves with respect to the Fubini-Study metric and the height of some toric bundles.
Toric Varieties
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Author : David A. Cox
language : en
Publisher: American Mathematical Society
Release Date : 2024-06-25
Toric Varieties written by David A. Cox and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-06-25 with Mathematics categories.
Toric varieties form a beautiful and accessible part of modern algebraic geometry. This book covers the standard topics in toric geometry; a novel feature is that each of the first nine chapters contains an introductory section on the necessary background material in algebraic geometry. Other topics covered include quotient constructions, vanishing theorems, equivariant cohomology, GIT quotients, the secondary fan, and the minimal model program for toric varieties. The subject lends itself to rich examples reflected in the 134 illustrations included in the text. The book also explores connections with commutative algebra and polyhedral geometry, treating both polytopes and their unbounded cousins, polyhedra. There are appendices on the history of toric varieties and the computational tools available to investigate nontrivial examples in toric geometry. Readers of this book should be familiar with the material covered in basic graduate courses in algebra and topology, and to a somewhat lesser degree, complex analysis. In addition, the authors assume that the reader has had some previous experience with algebraic geometry at an advanced undergraduate level. The book will be a useful reference for graduate students and researchers who are interested in algebraic geometry, polyhedral geometry, and toric varieties.
Convex Bodies And Algebraic Geometry
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Author : Tadao Oda
language : en
Publisher: Springer Verlag
Release Date : 1988
Convex Bodies And Algebraic Geometry written by Tadao Oda and has been published by Springer Verlag this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988 with Mathematics categories.
Geometry Of Toric Varieties
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Author : Laurent Bonavero
language : en
Publisher:
Release Date : 2002
Geometry Of Toric Varieties written by Laurent Bonavero and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Algebraic varieties categories.
Toric varieties form a beautiful class of algebraic varieties, which are often used as a testing ground for verifying general conjectures in algebraic geometry, for example, in Hilbert schemes, singularity theory, Mori theory, and so on. This volume gathers expanded versions of lectures presented during the summer school of ``Geometry of Toric Varieties'' in Grenoble (France). These lectures were given during the second and third weeks of the school. (The first week was devoted to introductory material.) The paper by D. Cox is an overview of recent work in toric varieties and its applications, putting the other contributions of the volume into perspective.
Introduction To Toric Varieties
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Author : William Fulton
language : en
Publisher: Princeton University Press
Release Date : 1993
Introduction To Toric Varieties written by William Fulton and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993 with Mathematics categories.
Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories. The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.
Arithmetic Geometry
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Author : Clay Mathematics Institute. Summer School
language : en
Publisher: American Mathematical Soc.
Release Date : 2009
Arithmetic Geometry written by Clay Mathematics Institute. Summer School and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Mathematics categories.
Based on survey lectures given at the 2006 Clay Summer School on Arithmetic Geometry at the Mathematics Institute of the University of Gottingen, this tile is intended for graduate students and recent PhD's. It introduces readers to modern techniques and conjectures at the interface of number theory and algebraic geometry.
Arithmetic Geometry Cryptography And Coding Theory
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Author : Alp Bassa
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-03-27
Arithmetic Geometry Cryptography And Coding Theory written by Alp Bassa and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-27 with Computers categories.
This volume contains the proceedings of the 15th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory (AGCT), held at the Centre International de Rencontres Mathématiques in Marseille, France, from May 18–22, 2015. Since the first meeting almost 30 years ago, the biennial AGCT meetings have been one of the main events bringing together researchers interested in explicit aspects of arithmetic geometry and applications to coding theory and cryptography. This volume contains original research articles reflecting recent developments in the field.
Tropical Geometry And Mirror Symmetry
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Author : Mark Gross
language : en
Publisher: American Mathematical Soc.
Release Date : 2011-01-20
Tropical Geometry And Mirror Symmetry written by Mark Gross and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-01-20 with Mathematics categories.
Tropical geometry provides an explanation for the remarkable power of mirror symmetry to connect complex and symplectic geometry. The main theme of this book is the interplay between tropical geometry and mirror symmetry, culminating in a description of the recent work of Gross and Siebert using log geometry to understand how the tropical world relates the A- and B-models in mirror symmetry. The text starts with a detailed introduction to the notions of tropical curves and manifolds, and then gives a thorough description of both sides of mirror symmetry for projective space, bringing together material which so far can only be found scattered throughout the literature. Next follows an introduction to the log geometry of Fontaine-Illusie and Kato, as needed for Nishinou and Siebert's proof of Mikhalkin's tropical curve counting formulas. This latter proof is given in the fourth chapter. The fifth chapter considers the mirror, B-model side, giving recent results of the author showing how tropical geometry can be used to evaluate the oscillatory integrals appearing. The final chapter surveys reconstruction results of the author and Siebert for ``integral tropical manifolds.'' A complete version of the argument is given in two dimensions.