Algebraic geometry Books

Book SynopsisThis volume contains articles related to the work of the Simons Collaboration “Arithmetic Geometry, Number Theory, and Computation.” The papers present mathematical results and algorithms necessary for the development of large-scale databases like the L-functions and Modular Forms Database (LMFDB). The authors aim to develop systematic tools for analyzing Diophantine properties of curves, surfaces, and abelian varieties over number fields and finite fields. The articles also explore examples important for future research.Specific topics include● algebraic varieties over finite fields● the Chabauty-Coleman method● modular forms● rational points on curves of small genus● S-unit equations and integral points.Table of Contents A robust implementation for solving the S-unit equation and several application (C. Rasmussen).- Computing classical modular forms for arbitrary congruence subgroups (E. Assaf).- Square root time Coleman integration on superelliptic curves (A. Best).- Computing classical modular forms ( A. Sutherland).- Elliptic curves with good reduction outside of the first six primes (B. Matschke).- Efficient computation of BSD invariants in genus 2 (R. van Bommel).- Restrictions on Weil polynomials of Jacobians of hyperelliptic curves (E. Costa).- Zen and the art of database maintenance (D. Roe).- Effective obstructions to lifting Tate classes from positive characteristic (E. Costa).- Conjecture: 100% of elliptic surfaces over Q have rank zero (A. Cowan).- On rational Bianchi newforms and abelian surfaces with quaternionic multiplication (J. Voight).- A database of Hilbert modular forms (J. Voight).- Isogeny classes of Abelian Varieties over Finite Fields in the LMFDB (D. Roe).- Computing rational points on genus 3 hyperelliptic curves (S. Hashimoto).- Curves with sharp Chabauty-Coleman bound (S. Gajović).- Chabauty-Coleman computations on rank 1 Picard curves (S. Hashimoto).- Linear dependence among Hecke eigenvalues (D. Kim).- Congruent number triangles with the same hypotenuse (D. Lowry-Duda).- Visualizing modular forms (D. Lowry-Duda).- A Prym variety with everywhere good reduction over Q(√ 61) ( J. Voight).- The S-integral points on the projective line minus three points via étale covers and Skolem's method (B. Poonen).

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Book SynopsisThe chapters in this volume explore the influence of the Russian school on the development of algebraic geometry and representation theory, particularly the pioneering work of two of its illustrious members, Alexander Beilinson and Victor Ginzburg, in celebration of their 60th birthdays. Based on the work of speakers and invited participants at the conference “Interactions Between Representation Theory and Algebraic Geometry”, held at the University of Chicago, August 21-25, 2017, this volume illustrates the impact of their research and how it has shaped the development of various branches of mathematics through the use of D-modules, the affine Grassmannian, symplectic algebraic geometry, and other topics. All authors have been deeply influenced by their ideas and present here cutting-edge developments on modern topics. Chapters are organized around three distinct themes: Groups, algebras, categories, and representation theory D-modules and perverse sheaves Analogous varieties defined by quivers Representation Theory and Algebraic Geometry will be an ideal resource for researchers who work in the area, particularly those interested in exploring the impact of the Russian school.Table of ContentsPart I: Groups, algebras, categories, and their representation theory.- On semisimplification of tensor categories.- Total aspherical parameters for Cherednik algebras.- Microlocal approach to Lusztig's symmetries.- Part II: D-modules and perverse sheaves, particularly on flag varieties and their generalizations.- Fourier-Sato Transform on hyperplane arrangements.- A quasi-coherent description of the category D-mod(Gr GL(n)).- The semi-infinite intersection cohomology sheaf--II: the Ran space version.- A topological approach to Soergel theory.- Part III: Varieties associated to quivers and relations to representation theory and symplectic geometry.- Loop Grassmannians of quivers and affine quantum groups.- Symplectic resolutions for multiplicative quiver varieties and character varieties for punctured surfaces.

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Book SynopsisThis book studies a class of monopoles defined by certain mild conditions, called periodic monopoles of generalized Cherkis–Kapustin (GCK) type. It presents a classification of the latter in terms of difference modules with parabolic structure, revealing a kind of Kobayashi–Hitchin correspondence between differential geometric objects and algebraic objects. It also clarifies the asymptotic behaviour of these monopoles around infinity.The theory of periodic monopoles of GCK type has applications to Yang–Mills theory in differential geometry and to the study of difference modules in dynamical algebraic geometry. A complete account of the theory is given, including major generalizations of results due to Charbonneau, Cherkis, Hurtubise, Kapustin, and others, and a new and original generalization of the nonabelian Hodge correspondence first studied by Corlette, Donaldson, Hitchin and Simpson.This work will be of interest to graduate students and researchers in differential and algebraic geometry, as well as in mathematical physics.Table of Contents. - Introduction. - Preliminaries. - Formal Difference Modules and Good Parabolic Structure. - Filtered Bundles. - Basic Examples of Monopoles Around Infinity. - Asymptotic Behaviour of Periodic Monopoles Around Infinity. - The Filtered Bundles Associated with Periodic Monopoles. - Global Periodic Monopoles of Rank One. - Global Periodic Monopoles and Filtered Difference Modules. - Asymptotic Harmonic Bundles and Asymptotic Doubly Periodic Instantons (Appendix).

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Book SynopsisThis volume is dedicated to Ciro Ciliberto on the occasion of his 70th birthday and contains refereed papers, offering an overview of important parts of current research in algebraic geometry and related research in the history of mathematics. It presents original research as well as surveys, both providing a valuable overview of the current state of the art of the covered topics and reflecting the versatility of the scientific interests of Ciro Ciliberto.Table of ContentsM. C. Brambilla, O. Dumitrescu, E. Postinghel, “Weyl cycles on the blow-up of $P^4$ at eight points".- A. Brigaglia, “Simson’s reconstruction of Apollonius’ Loci Plani. Modern ideas in classical language”.- F. Catanese, “Kummer quartic surfaces, strict self-duality, and more”.- L. Chiantini e Giorgio Ottaviani, “A footnote to a footnote to a paper of B. Segre”.- T. Dedieu and E. Sernesi, “Deformations and extensions of Gorenstein weighted projective spaces”.- V. Di Gennaro and Davide Franco, “Intersection cohomology and Severi Varieties”.- O. Dumitrescu and R. Miranda, “Cremona Orbits in $\mathbb P^4$ and Applications”.- F. Flamini and P. Supino, “On some components of Hilbert schemes of curves”.- Gerard van der Geer, “Siegel modular forms of degree two and three and invariant theory”.- A. Laface and L. Ugaglia, “On intrinsic negative curves”.- Angelo F. Lopez, with an appendix by Thomas Dedieu, “On the extendibility of projective varieties: a survey”.- M. Mella, “The minimal Cremona degree of quartic surfaces”.- M. Mendes Lopes and R. Pardini, “On the degree of the canonical map of a surface of general type”.- C. Pedrini, “Hyperkæhler varieties with a motive of abelian type”.- F. Polizzi and P. Sabatino, “Finite quotients of surface braid groups and double Kodaira fibrations”.- Y. Prokhorov and M. Zaidenberg, “Affine cones over Fano-Mukai fourfolds of genus 10 are flexible”.- J. Roé, “Enriques diagrams under pullback by a double cover”.- E. Rogora, “The “projective spirit” in Segre’s lectures on differential equations”.

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Book SynopsisThis volume celebrates the 100th birthday of Professor Chen-Ning Frank Yang (Nobel 1957), one of the giants of modern science and a living legend. Starting with reminiscences of Yang's time at the research centre for theoretical physics at Stonybrook (now named C. N. Yang Institute) by his successor Peter van Nieuwenhuizen, the book is a collection of articles by world-renowned mathematicians and theoretical physicists. This emphasizes the Dialogue Between Physics and Mathematics that has been a central theme of Professor Yang’s contributions to contemporary science. Fittingly, the contributions to this volume range from experimental physics to pure mathematics, via mathematical physics. On the physics side, the contributions are from Sir Anthony Leggett (Nobel 2003), Jian-Wei Pan (Willis E. Lamb Award 2018), Alexander Polyakov (Breakthrough Prize 2013), Gerard 't Hooft (Nobel 1999), Frank Wilczek (Nobel 2004), Qikun Xue (Fritz London Prize 2020), and Zhongxian Zhao (Bernd T. Matthias Prize 2015), covering an array of topics from superconductivity to the foundations of quantum mechanics. In mathematical physics there are contributions by Sir Roger Penrose (Nobel 2022) and Edward Witten (Fields Medal 1990) on quantum twistors and quantum field theory, respectively. On the mathematics side, the contributions by Vladimir Drinfeld (Fields Medal 1990), Louis Kauffman (Wiener Gold Medal 2014), and Yuri Manin (Cantor Medal 2002) offer novel ideas from knot theory to arithmetic geometry.Inspired by the original ideas of C. N. Yang, this unique collection of papers b masters of physics and mathematics provides, at the highest level, contemporary research directions for graduate students and experts alike.Table of Contents1 Frank Yang at Stony Brook and the Beginning of Supergravity.- 2. A Stacky Approach to Crystals.- 3 The Potts Model, the Jones Polynomial and Link Homology.- 4 The Penrose–Onsager–Yang Approach to Superconductivity and Superfluidity.- 5 Quantum Operads.- 6 Quantum computational complexity withphotons and linear optics.- 7 Quantized Twistors, G2*, and the Split Octonions.- 8 Kronecker Anomalies and Gravitational Striction.- 9 Projecting Local and Global Symmetries to the Planck Scale.- 10 Gauge Symmetry in Shape Dynamics.- 11 Why Does Quantum Field Theory In Curved Spacetime Make Sense? And What Happens To The Algebra of Observables In The Thermodynamic Limit?.- 12 Quantum Anomalous Hall Effect.- 13 Magic Superconducting States in Cuprates.

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Book SynopsisThis book collects the proceedings of a series of conferences dedicated to birational geometry of Fano varieties held in Moscow, Shanghai and PohangThe conferences were focused on the following two related problems:• existence of Kähler–Einstein metrics on Fano varieties• degenerations of Fano varietieson which two famous conjectures were recently proved. The first is the famous Borisov–Alexeev–Borisov Conjecture on the boundedness of Fano varieties, proved by Caucher Birkar (for which he was awarded the Fields medal in 2018), and the second one is the (arguably even more famous) Tian–Yau–Donaldson Conjecture on the existence of Kähler–Einstein metrics on (smooth) Fano varieties and K-stability, which was proved by Xiuxiong Chen, Sir Simon Donaldson and Song Sun. The solutions for these longstanding conjectures have opened new directions in birational and Kähler geometries. These research directions generated new interesting mathematical problems, attracting the attention of mathematicians worldwide.These conferences brought together top researchers in both fields (birational geometry and complex geometry) to solve some of these problems and understand the relations between them. The result of this activity is collected in this book, which contains contributions by sixty nine mathematicians, who contributed forty three research and survey papers to this volume. Many of them were participants of the Moscow–Shanghai–Pohang conferences, while the others helped to expand the research breadth of the volume—the diversity of their contributions reflects the vitality of modern Algebraic Geometry.Table of ContentsT. Abe, Classification of exceptional complements: elliptic curve case.- E. Ballico, E. Gasparim, F. Rubilar, B. Suzuki, LAGRANGIAN SKELETA, COLLARS AND DUALITY.- G. Belousov, CYLINDERS IN DEL PEZZO SURFACES OF DEGREE TWO.- M. Benzerga, FINITENESS OF REAL STRUCTURES ON KLT CALABI-YAU REGULAR SMOOTH PAIRS OF DIMENSION 2.- C. Birkar, ANTICANONICAL VOLUME OF FANO 4-FOLDS.- C. Boyer Christina Tonnesen-Friedman, CONSTANT SCALAR CURVATURE SASAKI METRICS AND PROJECTIVE BUNDLES.- G. Brown, J. Buczynski, A. Kasprzyk, TORIC SARKISOV LINKS.- I. Burban, DU VAL SINGULARITIES.- I. Cheltsov, H. Suess, K-POLYSTABILITY OF TWO SMOOTH FANO THREEFOLDS.- G. Codogni, Z. Patakfalvi, A NOTE ON FAMILIES OF K-SEMISTABLE LOG-FANO PAIRS.- T. Delcroix, THE YAU-TIAN-DONALDSON CONJECTURE FOR COHOMOGENEITY ONE MANIFOLDS.- A. Dubouloz, FIBRATIONS BY AFFINE LINES ON RATIONAL AFFINE SURFACES WITH IRREDUCIBLE BOUNDARIES.- K. Fujita, ON FANO THREEFOLDS OF DEGREE 22 AFTER CHELTSOV AND SHRAMOV.- K. Fujita, Y. Liu, H. Suess, K. Zhang, Z. Zhuang, ON THE CHELTSOV-RUBINSTEIN CONJECTURE.- S. Grishin, Ilya Karzhemanov, Ming-Chang Kang, RATIONALITY OF QUOTIENTS BY FINITE HEISENBERG GROUPS.- Y. Hashimoto.- J. Keller, QUOT-SCHEME LIMIT OF FUBINI–STUDY METRICS AND ITS APPLICATIONS TO BALANCED METRICS.- Z. Hu, EXISTENCE OF CANONICAL MODELS FOR KAWAMATA LOG TERMINAL PAIRS.- Y. Imagi, GENERALIZED THOMAS–YAU UNIQUENESS THEOREMS.- K. Jamieson, BIRATIONALLY RIGID COMPLETE INTERSECTIONS OF CODIMENSION 3.- D. Jeong.- J. Park, SIMPLY CONNECTED SASAKI-EINSTEIN 5-MANIFOLDS: OLD AND NEW.- C. Jiang, CHARACTERIZING Q-FANO THREEFOLDS WITH THE SMALLEST ANTI-CANONICAL VOLUME.- L. Katzarkov, Kyoung-Seog Lee, J. Svoboda, A. Petkov, INTERPRETATIONS OF SPECTRA.- Young-Hoon Kiem, Kyoung-Seog Lee, FANO VISITORS, FANO DIMENSION AND FANO ORBIFOLDS.- In-kyun Kim, N. Viswanathan, J. Won, ON SINGULAR DEL PEZZO HYPERSURFACES OF INDEX 3.- S. Kudryavtsev, Blow-ups of three-dimensional toric singularities.- N. Kurnosov, E. Yasinsky, AUTOMORPHISMS OF HYPERKAHLER MANIFOLDS AND GROUPS ACTING ON CAT(0) SPACES.- A. Laface, R. Quezada, ON GENERALIZED BUCHI SURFACES.- Chi Li, K-STABILITY AND FUJITA APPROXIMATION.- Y. Li, Zhenye Li, ON A CONJECTURE OF FULTON ON ISOTROPIC GRASSMANNIANS.- Y. Maeda, Y. Odaka, FANO SHIMURA VARIETIES WITH MOSTLY BRANCHED CUSP.- L. Makar-Limanov, ON LOCALLY NILPOTENT DERIVATIONS OF DANIELEWSKI DOMAINS.- D. Markouchevitch, A. Moreau, ACTION OF THE AUTOMORPHISM GROUP ON THE JACOBIAN OF KLEIN'S QUARTIC CURVE.- J. Martinez-Garcia, C. Spotti, SOME OBSERVATIONS ON THE DIMENSION OF FANO K-MODULI.- D. Witt Nystrom, OKOUNKOV BODIES AND THE KAHLER GEOMETRY OF PROJECTIVE MANIFOLDS.- J. Park, SINGULARITIES OF PLURI-FUNDAMENTAL DIVISORS ON GORENSTEIN FANO VARIETIES OF COINDEX.- J. Paulhus, A DATABASE OF GROUP ACTIONS ON RIEMANN SURFACES.- A. Petracci, A 1-DIMENSIONAL COMPONENT OF K-MODULI OF DEL PEZZO SURFACES.- T. De Piro, A NON-STANDARD BEZOUT THEOREM FOR CURVES.- Y. Prokhorov, EMBEDDINGS OF THE SYMMETRIC GROUPS TO THE SPACE CREMONA GROUP.- J. Ross, M. Toma, ON HODGE-RIEMANN COHOMOLOGY CLASSES.- Y. Rubinstein, ON LARGE DEVIATION PRINCIPLES AND THE MONGE–AMPERE EQUATION (FOLLOWING BERMAN, HULTGREN).- T. Sano, ON BIRATIONAL BOUNDEDNESS OF SOME CALABI-YAU HYPERSURFACES.- Y. Zarhin, ABELIAN VARIETIES, QUATERNION TRICK AND ENDOMORPHISMS.

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Book SynopsisThis textbook offers an introduction to abelian varieties, a rich topic of central importance to algebraic geometry. The emphasis is on geometric constructions over the complex numbers, notably the construction of important classes of abelian varieties and their algebraic cycles.The book begins with complex tori and their line bundles (theta functions), naturally leading to the definition of abelian varieties. After establishing basic properties, the moduli space of abelian varieties is introduced and studied. The next chapters are devoted to the study of the main examples of abelian varieties: Jacobian varieties, abelian surfaces, Albanese and Picard varieties, Prym varieties, and intermediate Jacobians. Subsequently, the Fourier–Mukai transform is introduced and applied to the study of sheaves, and results on Chow groups and the Hodge conjecture are obtained.This book is suitable for use as the main text for a first course on abelian varieties, for instance as a second graduate course in algebraic geometry. The variety of topics and abundant exercises also make it well suited to reading courses. The book provides an accessible reference, not only for students specializing in algebraic geometry but also in related subjects such as number theory, cryptography, mathematical physics, and integrable systems.Trade Review“The reorganization of the topics is fine surgical work. Several portions of the original monograph are sewn in a natural way in the new book, adding examples or additional text when necessary, and re-arranging the focus to make it a more friendly introduction to the subject. Careful attention to details and the required background makes the book under review accessible to an interested reader and could be a used as textbook for a course on abelian varieties.” (Felipe Zaldivar, MAA Reviews, June 18, 2023)Table of Contents1. Line Bundles on Complex Tori.- 2 Abelian Varieties.- 3 Moduli Spaces.- 4 Jacobian Varieties.- 5 Main Examples of Abelian Varieties.- 6 The Fourier Transform for Sheaves and Cycles.- 7 Introduction to the Hodge Conjecture for Abelian Varieties.

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Book SynopsisThis book provides an elementary introduction, complete with detailed proofs, to the celebrated tilings of the plane discovered by Sir Roger Penrose in the '70s. Quasi-periodic tilings of the plane, of which Penrose tilings are the most famous example, started as recreational mathematics and soon attracted the interest of scientists for their possible application in the description of quasi-crystals. The purpose of this survey, illustrated with more than 200 figures, is to introduce the curious reader to this beautiful topic and be a reference for some proofs that are not easy to find in the literature. The volume covers many aspects of Penrose tilings, including the study, from the point of view of Connes' Noncommutative Geometry, of the space parameterizing these tilings.Table of ContentsIntroduction.- Tilings and puzzles.- Robinson triangles.- Penrose tilings.- De Bruijn’s pentagrids.- The noncommutative space of Penrose tilings.-Some useful formulas.

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Book SynopsisThis book constitutes the proceedings of the Workshop Empowering Novel Geometric Algebra for Graphics and Engineering, ENGAGE 2022, held in conjunction with Computer Graphics International conference, CGI 2022, which took place virtually, in September 2022. The 10 full papers included in this volume were carefully reviewed and selected from 12 submissions. The workshop focused specifically on important aspects of geometric algebra including algebraic foundations, digitized transformations, orientation, conic fitting, protein modelling, digital twinning, and multidimensional signal processing.Table of ContentsFoundations of Geometric Algebra.- Transformations, Orientation and Fitting.- Modelling Proteins and Cities.- Signal Processing with Octonions.

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Book SynopsisSemi-Infinite Geometry is a theory of "doubly infinite-dimensional" geometric or topological objects. In this book the author explains what should be meant by an algebraic variety of semi-infinite nature. Then he applies the framework of semiderived categories, suggested in his previous monograph titled Homological Algebra of Semimodules and Semicontramodules, (Birkhäuser, 2010), to the study of semi-infinite algebraic varieties. Quasi-coherent torsion sheaves and flat pro-quasi-coherent pro-sheaves on ind-schemes are discussed at length in this book, making it suitable for use as an introduction to the theory of quasi-coherent sheaves on ind-schemes. The main output of the homological theory developed in this monograph is the functor of semitensor product on the semiderived category of quasi-coherent torsion sheaves, endowing the semiderived category with the structure of a tensor triangulated category. The author offers two equivalent constructions of the semitensor product, as well as its particular case, the cotensor product, and shows that they enjoy good invariance properties. Several geometric examples are discussed in detail in the book, including the cotangent bundle to an infinite-dimensional projective space, the universal fibration of quadratic cones, and the important popular example of the loop group of an affine algebraic group.Table of Contents- 1. Ind-Schemes and Their Morphisms. - 2. Quasi-Coherent Torsion Sheaves. - 3. Flat Pro-Quasi-Coherent Pro-Sheaves. - 4. Dualizing Complexes on Ind-Noetherian Ind-Schemes. - 5. The Cotensor Product. - 6. Ind-Schemes of Ind-Finite Type and the factorial !-Tensor Product. - 7. X-Flat Pro-Quasi-Coherent Pro-Sheaves on Y. - 8. The Semitensor Product. - 9. Flat Affine Ind-Schemes over Ind-Schemes of Ind-Finite Type. - 10. Invariance Under Postcomposition with a Smooth Morphism. - 11. Some Infinite-Dimensional Geometric Examples.

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Book Synopsis- Introduction.- Koszul duality equivalence.- Categorical DT theory for local surfaces.- D-critical D/K equivalence conjectures.- Categorical wall-crossing via Koszul duality.- Window theorem for DT categories.- Categori ed Hall products on DT categories.- Some auxiliary results.

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Book SynopsisSingularities of bridgeland moduli spaces for k3 categories: an update.- On rigid manifolds of kodaira dimension 1.- On the components of the main stream of the moduli space of surfaces of general type with pg = q = 2.- Degree of irrationality of fano threefold hypersurfaces.- Non effective planar linear systems at the boundary of the mori cone.- Sur l'injectivite de l'application cycle de jannsen.- A prym hypergeometric.- Complete curves in the moduli space of polarized k3 surfaces and hyper-kahler manifolds.- The grothendieck group of algebraic stacks.- Conformal blocks in genus zero and the kz connection.- Second fundamental form and higher gaussian maps.- On infinitesimal invariants of normal functions.- Macaulay duality and its geometry.- On the chow ring of fano fourfolds of k3 type.- Quadratic counts of twisted cubics.

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Book SynopsisOn varieties with ulrich twisted normal bundles.- Rescalability of integrable mixed twistor d-modules.- Cohomology of complete intersections of quadrics.- Generic torelli for coverings of plane quintics ramified in two points.- Theta groups and projective models of hyperkahler varieties.- Footnotes to the birational geometry of primitive symplectic varieties.- Finitude uniforme pour les cycles de codimension 2 sur les corps de nombres.- The hesse pencil and polarizations of type (1,3) on abelian surfaces.- Enriched hodge structures and cycles on complex analytic thickenings.- Notes on the cohomology of local systems of small weight on m_2.- Burnside groups and orbifold invariants of birational maps.- Enumerative geometry of legendrian foliations: a tale of contact.- Geometric representability of 1-cycles on rationally connected threefolds.- Module structure of the k-theory of polynomial-like rings.

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Book Synopsis1 Bill Bruce, Peter Giblin, David Mond, Stephen Pizer and Les Wilson, Jim Damon's Contributions to Singularity Theory and Its Applications.- 2 Viktor A. Vassiliev, Real Function Singularities and Their Bifurcation Sets.- 3 Adam Parusinski and Armin Rainer, Perturbation Theory of Polynomials and Linear Operators.- 4 Goo Ishikawa, Frontal Singularities and Related Problems.- 5 Osamu Saeki, Introduction to Global Singularity Theory of Differentiable Maps.- 6 Claude Sabbah, Singularities of Functions: A Global Point of View.- 7 Mark McLean, Floer Theory, Arc Spaces and Singularities.- 8 Stephen S.-T. Yau and Huaiqing Zuo, Various Derivation Lie Algebras of Isolated Singularities.- 9 Bingyi Chen, Stephen S.-T. Yau and Huaiqing Zuo, Three-Dimensional Rational Isolated Complete Intersection Singularities.- 10 Ziquan Zhuang, Stability of klt Singularities.- 11 Qianyu Chen, Bradley Dirks and Mircea Mustata, An introduction to V-Filtration.- 12 Kiyoshi Takeuchi, Geometric Monodromies, Mixed Hodge Numbers of Motivic Milnor Fibers and Newton Polyhedra.- 13 Willem Veys, Introduction to the Monodromy Conjecture.- 14 Laurentiu G. Maxim, Jose Israel Rodriguez and Botong Wang, Applications of Singularity Theory in Applied Algebraic Geometry and Algebraic Statistics.

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Book SynopsisThis volume collects the texts of five courses given in the Arithmetic Geometry Research Programme 2009-2010 at the CRM Barcelona. All of them deal with characteristic p global fields; the common theme around which they are centered is the arithmetic of L-functions (and other special functions), investigated in various aspects. Three courses examine some of the most important recent ideas in the positive characteristic theory discovered by Goss (a field in tumultuous development, which is seeing a number of spectacular advances): they cover respectively crystals over function fields (with a number of applications to L-functions of t-motives), gamma and zeta functions in characteristic p, and the binomial theorem. The other two are focused on topics closer to the classical theory of abelian varieties over number fields: they give respectively a thorough introduction to the arithmetic of Jacobians over function fields (including the current status of the BSD conjecture and its geometric analogues, and the construction of Mordell-Weil groups of high rank) and a state of the art survey of Geometric Iwasawa Theory explaining the recent proofs of various versions of the Main Conjecture, in the commutative and non-commutative settings.Table of ContentsCohomological Theory of Crystals over Function Fields and Applications.- On Geometric Iwasawa Theory and Special Values of Zeta Functions.- The Ongoing Binomial Revolution.- Arithmetic of Gamma, Zeta and Multizeta Values for Function Fields.- Curves and Jacobians over Function Fields.

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Book SynopsisThe contributions in this book explore various contexts in which the derived category of coherent sheaves on a variety determines some of its arithmetic. This setting provides new geometric tools for interpreting elements of the Brauer group. With a view towards future arithmetic applications, the book extends a number of powerful tools for analyzing rational points on elliptic curves, e.g., isogenies among curves, torsion points, modular curves, and the resulting descent techniques, as well as higher-dimensional varieties like K3 surfaces. Inspired by the rapid recent advances in our understanding of K3 surfaces, the book is intended to foster cross-pollination between the fields of complex algebraic geometry and number theory.Contributors:· Nicolas Addington · Benjamin Antieau · Kenneth Ascher · Asher Auel · Fedor Bogomolov · Jean-Louis Colliot-Thélène · Krishna Dasaratha · Brendan Hassett · Colin Ingalls · Martí Lahoz · Emanuele Macrì · Kelly McKinnie · Andrew Obus · Ekin Ozman · Raman Parimala · Alexander Perry · Alena Pirutka · Justin Sawon · Alexei N. Skorobogatov · Paolo Stellari · Sho Tanimoto · Hugh Thomas · Yuri Tschinkel · Anthony Várilly-Alvarado · Bianca Viray · Rong ZhouTable of ContentsThe Brauer group is not a derived invariant.- Twisted derived equivalences for affine schemes.- Rational points on twisted K3 surfaces and derived equivalences.- Universal unramified cohomology of cubic fourfolds containing a plane.- Universal spaces for unramified Galois cohomology.- Rational points on K3 surfaces and derived equivalence.- Unramified Brauer classes on cyclic covers of the projective plane.- Arithmetically Cohen-Macaulay bundles on cubic fourfolds containing a plane.- Brauer groups on K3 surfaces and arithmetic applications.- On a local-global principle for H3 of function fields of surfaces over a finite field.- Cohomology and the Brauer group of double covers.

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Book SynopsisThis is a two-volume collection presenting the selected works of Herbert Busemann, one of the leading geometers of the twentieth century and one of the main founders of metric geometry, convexity theory and convexity in metric spaces. Busemann also did substantial work (probably the most important) on Hilbert’s Problem IV. These collected works include Busemann’s most important published articles on these topics. Volume I of the collection features Busemann’s papers on the foundations of geodesic spaces and on the metric geometry of Finsler spaces. Volume II includes Busemann’s papers on convexity and integral geometry, on Hilbert’s Problem IV, and other papers on miscellaneous subjects. Each volume offers biographical documents and introductory essays on Busemann’s work, documents from his correspondence and introductory essays written by leading specialists on Busemann’s work. They are a valuable resource for researchers in synthetic and metric geometry, convexity theory and the foundations of geometry. Table of ContentsPreface.- Introduction to Volume I.- List of publications of Herbert Busemann.- Acknowledgements.- Essays.- A. Papadpoulos: Herbert Busemann (1905-1994).- A. Papadopoulos and M. Troyanov: On three early papers by Herbert Busemann on the foundations of geometry.- M. Troyanov: On Pasch's Axiom and Desargues' Theorem in Busemann's work.- V. N. Berestovskiy: Busemann's results, ideas, questions on locally compact homogeneous geodesic spaces.- A. Papadopoulos and S. Yamada: Busemann's problems on G-spaces.- Busemann's metric theory of timelike spaces.- A. Papadopoulos: Chronogeometry.- W. M. Boothby: Review of Busemann's book The geometry of Geodesics.- F. A. Ficken: Review of Busemann's book Metric Methods in Finsler Spaces and in the Foundations of Geometry.- Busemann's papers on the foundations of geodesic spaces and on the metric geometry of Finsler spaces.

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Book SynopsisThis two volume work on Positivity in Algebraic Geometry contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. A good deal of this material has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments. Volume I is more elementary than Volume II, and, for the most part, it can be read without access to Volume II. Table of ContentsNotation and Conventions.- One: Ample Line Bundles and Linear Series.- to Part One.- 1 Ample and Nef Line Bundles.- 2 Linear Series.- 3 Geometric Manifestations of Positivity.- 4 Vanishing Theorems.- 5 Local Positivity.- Appendices.- A Projective Bundles.- B Cohomology and Complexes.- B.1 Cohomology.- B.2 Complexes.- References.- Glossary of Notation.

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Book SynopsisTwo volume work containing a contemporary account on "Positivity in Algebraic Geometry". Both volumes also available as hardcover editions as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete". A good deal of the material has not previously appeared in book form. Volume II is more at the research level and somewhat more specialized than Volume I. Volume II contains a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. Contains many concrete examples, applications, and pointers to further developmentsTrade ReviewFrom the reviews: "The main theme of this ... monograph is a comprehensive description of the fields of complex algebraic geometry connected with the notion of positivity. ... The book is written for mathematicians interested in the modern development of algebraic geometry." (EMS Newsletter, September, 2006)Table of ContentsNotation and Conventions.- Two: Positivity for Vector Bundles.- 6 Ample and Nef Vector Bundles.- 6.1 Classical Theory.- 6.1.A Definition and First Properties.- 6.1.B Cohomological Properties.- 6.1.C Criteria for Amplitude.- 6.1.D Metric Approaches to Positivity of Vector Bundles.- 6.2 Q-Twisted and Nef Bundles.- 6.2.A Twists by Q-Divisors.- 6.2.B Nef Bundles.- 6.3 Examples and Constructions.- 6.3.A Normal and Tangent Bundles.- 6.3.B Ample Cotangent Bundles and Hyperbolicity.- 6.3.C Picard Bundles.- 6.3.D The Bundle Associated to a Branched Covering.- 6.3.E Direct Images of Canonical Bundles.- 6.3.F Some Constructions of Positive Vector Bundles.- 6.4 Ample Vector Bundles on Curves.- 6.4.A Review of Semistability.- 6.4.B Semistability and Amplitude.- Notes.- 7 Geometric Properties of Ample Bundles.- 7.1 Topology.- 7.1.A Sommese’s Theorem.- 7.1.B Theorem of Bloch and Gieseker.- 7.1.C A Barth-Type Theorem for Branched Coverings.- 7.2 Degeneracy Loci.- 7.2.A Statements and First Examples.- 7.2.B Proof of Connectedness of Degeneracy Loci.- 7.2.C Some Applications.- 7.2.D Variants and Extensions.- 7.3 Vanishing Theorems.- 7.3.A Vanishing Theorems of Griffiths and Le Potier.- 7.3.B Generalizations.- Notes.- 8 Numerical Properties of Ample Bundles.- 8.1 Preliminaries from Intersection Theory.- 8.1.A Chern Classes for Q-Twisted Bundles.- 8.1.B Cone Classes.- 8.1.C Cone Classes for Q-Twists.- 8.2 Positivity Theorems.- 8.2.A Positivity of Chern Classes.- 8.2.B Positivity of Cone Classes.- 8.3 Positive Polynomials for Ample Bundles.- 8.4 Some Applications.- 8.4.A Positivity of Intersection Products.- 8.4.B Non-Emptiness of Degeneracy Loci.- 8.4.C Singularities of Hypersurfaces Along a Curve.- Notes.- Three: Multiplier Ideals and Their Applications.- 9 Multiplier Ideal Sheaves.- 9.1 Preliminaries.- 9.1.A Q-Divisors.- 9.1.B Normal Crossing Divisors and Log Resolutions.- 9.1.C The Kawamata—Viehweg Vanishing Theorem.- 9.2 Definition and First Properties.- 9.2.A Definition of Multiplier Ideals.- 9.2.B First Properties.- 9.3 Examples and Complements.- 9.3.A Multiplier Ideals and Multiplicity.- 9.3.B Invariants Arising from Multiplier Ideals.- 9.3.C Monomial Ideals.- 9.3.D Analytic Construction of Multiplier Ideals.- 9.3.E Adjoint Ideals.- 9.3.F Multiplier and Jacobian Ideals.- 9.3.G Multiplier Ideals on Singular Varieties.- 9.4 Vanishing Theorems for Multiplier Ideals.- 9.4.A Local Vanishing for Multiplier Ideals.- 9.4.B The Nadel Vanishing Theorem.- 9.4.C Vanishing on Singular Varieties.- 9.4.D Nadel’s Theorem in the Analytic Setting.- 9.4.E Non-Vanishing and Global Generation.- 9.5 Geometric Properties of Multiplier Ideals.- 9.5.A Restrictions of Multiplier Ideals.- 9.5.B Subadditivity.- 9.5.C The Summation Theorem.- 9.5.D Multiplier Ideals in Families.- 9.5.E Coverings.- 9.6 Skoda’s Theorem.- 9.6.A Integral Closure of Ideals.- 9.6.B Skoda’s Theorem: Statements.- 9.6.C Skoda’s Theorem: Proofs.- 9.6.D Variants.- Notes.- 10 Some Applications of Multiplier Ideals.- 10.1 Singularities.- 10.1.A Singularities of Projective Hypersurfaces.- 10.1.B Singularities of Theta Divisors.- 10.1.C A Criterion for Separation of Jets of Adjoint Series.- 10.2 Matsusaka’s Theorem.- 10.3 Nakamaye’s Theorem on Base Loci.- 10.4 Global Generation of Adjoint Linear Series.- 10.4.A Fujita Conjecture and Angehrn—Siu Theorem.- 10.4.B Loci of Log-Canonical Singularities.- 10.4.C Proof of the Theorem of Angehrn and Siu.- 10.5 The Effective Nullstellensatz.- Notes.- 11 Asymptotic Constructions.- 11.1 Construction of Asymptotic Multiplier Ideals.- 11.1.A Complete Linear Series.- 11.1.B Graded Systems of Ideals and Linear Series.- 11.2 Properties of Asymptotic Multiplier Ideals.- 11.2.A Local Statements.- 11.2.B Global Results.- 11.2.C Multiplicativity of Plurigenera.- 11.3 Growth of Graded Families and Symbolic Powers.- 11.4 Fujita’s Approximation Theorem.- 11.4.A Statement and First Consequences.- 11.4.B Proof of Fujita’s Theorem.- 11.4.C The Dual of the Pseudoeffective Cone.- 11.5.- Notes.- References.- Glossary of Notation.

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Book SynopsisFrom the reviews: "Although several textbooks on modern algebraic geometry have been published in the meantime, Mumford's "Volume I" is, together with its predecessor the red book of varieties and schemes, now as before one of the most excellent and profound primers of modern algebraic geometry. Both books are just true classics!" Zentralblatt Trade Review"In the 20th century, algebraic geometry has undergone several revolutionary changes with respect to its conceptual foundations, technical framework, and intertwining with other branches of mathematics. Accordingly the way it is taught has gone through distinct phases. The theory of algebraic schemes, together with its full-blown machinery of sheaves and their cohomology, being for now the ultimate stage of this evolution process in algebraic geometry, had created -- around 1960 -- the urgent demand for new textbooks reflecting these developments and (henceforth) various facets of algebraic geometry. ... It was David Mumford, who at first started the project of writing a textbook on algebraic geometry in its new setting. His mimeographed Harvard notes ntroduction to algebraic geometry: Preliminary version of the first three chapters (bound in red) were distributed in the mid 1960's, and they were intended as the first stage of a forthcoming, more inclusive textbook. For some years, these mimeographed notes represented the almost only, however utmost convenient and abundant source for non-experts to get acquainted with the basic new concepts and ideas of modern algebraic geometry. Their timeless utility, in this regard, becomes apparent from the fact that two reprints of them have appeared, since 1988, as a proper book under the title he red book of varieties and schemes' ( Lect. Notes Math. 1358). In the process of exending his Harvard notes to a comprehensive textbook, the author's teaching experiences led him to the didactic conclusion that it would be better to split the book into two volumes, thereby starting with complex projective varieties (in volume I), and proceeding with schemes and their cohomology (in volume II). -- In 1976, the author published the first volume under the title lgebraic geometry. I: Complex projective varieties where the corrections concerned the wiping out of some misprints, inconsistent notations, and other slight inaccuracies.The book under review is an unchanged reprint of this corrected second edition from 1980. Although several textbooks on modern algebraic geometry have been published in the meantime, Mumford's "Volume I" is, together with its predecessor the red book of varieties and schemes now as before, one of the most excellent and profound primers of modern algebraic geometry. Both books are just true classics!"Zentralblatt MATH, 821Table of Contents1. Affine Varieties.- §1A. Their Definition, Tangent Space, Dimension, Smooth and Singular Points.- §1B. Analytic Uniformization at Smooth Points, Examples of Topological Knottedness at Singular Points.- §1C. Ox,X a UFD when x Smooth; Divisor of Zeroes and Poles of Functions.- 2. Projective Varieties.- §2A. Their Definition, Extension of Concepts from Affine to Projective Case.- §2B. Products, Segre Embedding, Correspondences.- §2C. Elimination Theory, Noether’s Normalization Lemma, Density of Zariski-Open Sets.- 3. Structure of Correspondences.- §3A. Local Properties—Smooth Maps, Fundamental Openness Principle, Zariski’s Main Theorem.- §3B. Global Properties—Zariski’s Connectedness Theorem, Specialization Principle.- §3C. Intersections on Smooth Varieties.- 4. Chow’s Theorem.- §4A. Internally and Externally Defined Analytic Sets and their Local Descriptions as Branched Coverings of ?n.- §4B. Applications to Uniqueness of Algebraic Structure and Connectedness.- 5. Degree of a Projective Variety.- §5A. Definition of deg X, multxX, of the Blow up Bx(X), Effect of a Projection, Examples.- §5B. Bezout’s Theorem.- §5C. Volume of a Projective Variety ; Review of Homology, DeRham’s Theorem, Varieties as Minimal Submanifolds.- 6. Linear Systems.- §6A. The Correspondence between Linear Systems and Rational Maps, Examples; Complete Linear Systems are Finite-Dimensional.- §6B. Differential Forms, Canonical Divisors and Branch Loci.- §6C. Hilbert Polynomials, Relations with Degree.- Appendix to Chapter 6. The Weil-Samuel Algebraic Theory of Multiplicity.- 7. Curves and Their Genus.- §7A. Existence and Uniqueness of the Non-Singular Model of Each Function Field of Transcendence Degree 1 (after Albanese).- §7B. Arithmetic Genus = Topological Genus; Existence of Good Projections to ?1, ?2, ?3.- §7C. Residues of Differentials on Curves, the Classical Riemann-Roch Theorem for Curves and Applications.- §7D. Curves of Genus 1 as Plane Cubics and as Complex Tori ?/L.- 8. The Birational Geometry of Surfaces.- §8A. Generalities on Blowing up Points.- §8B. Resolution of Singularities of Curves on a Smooth Surface by Blowing up the Surface; Examples.- §8C. Factorization of Birational Maps between Smooth Surfaces; the Trees of Infinitely Near Points.- §8D. The Birational Map between ?2 and the Quadric and Cubic Surfaces; the 27 Lines on a Cubic Surface.- List of Notations.

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Book SynopsisMumford's famous "Red Book" gives a simple, readable account of the basic objects of algebraic geometry, preserving as much as possible their geometric flavor and integrating this with the tools of commutative algebra. It is aimed at graduates or mathematicians in other fields wishing to quickly learn aboutalgebraic geometry. This new edition includes an appendix that gives an overview of the theory of curves, their moduli spaces and their Jacobians -- one of the most exciting fields within algebraic geometry.Trade Review"This is the second edition of a famous and well-known introduction to algebraic geometry, written to show that the language of schemes is fundamentally geometrical and clearly expressing the intuition of algebraic geometry. ... This book can strongly be recommended to anybody interested in algebraic geometry and willing to learn about varieties and schemes and their main problems."EMS Newsletter, Vol. 37, Sept. 2000Table of ContentsVarieties.- Preschemes.- Local properties of schemes.- References.- Appendix: Curves and their Jacobians.- Survey of work on the Schottky problem up to 1996 (by Enrico Arbarello).- References.- Guide to the Literature and References (Curves and Their Jacobians).- Supplementary Bibliography on the Schottky Problem.

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Book SynopsisThe section conjecture in anabelian geometry, announced by Grothendieck in 1983, is concerned with a description of the set of rational points of a hyperbolic algebraic curve over a number field in terms of the arithmetic of its fundamental group. While the conjecture is still open today in 2012, its study has revealed interesting arithmetic for curves and opened connections, for example, to the question whether the Brauer-Manin obstruction is the only one against rational points on curves. This monograph begins by laying the foundations for the space of sections of the fundamental group extension of an algebraic variety. Then, arithmetic assumptions on the base field are imposed and the local-to-global approach is studied in detail. The monograph concludes by discussing analogues of the section conjecture created by varying the base field or the type of variety, or by using a characteristic quotient or its birational analogue in lieu of the fundamental group extension.Trade ReviewFrom the book reviews:“The book under review, resulting from the author’s dissertation … is both a research monograph and a thorough presentation of the arithmetic and geometry of Grothendieck’s section conjecture from the foundations to the current state of the art. … It will be useful not only to specialists, as it is accessible to anyone familiar with the basics of modern algebraic geometry and the theory of algebraic fundamental groups.” (Marco A. Garuti, Mathematical Reviews, May, 2014)Table of ContentsPart I Foundations of Sections.- 1 Continuous Non-abelian H1 with Profinite Coefficients.-2 The Fundamental Groupoid.- 3 Basic Geometric Operations in Terms of Sections.- 4 The Space of Sections as a Topological Space.- 5 Evaluation of Units.- 6 Cycle Classes in Anabelian Geometry.- 7 Injectivity in the Section Conjecture.- Part II Basic Arithmetic of Sections.- 7 Injectivity in the Section Conjecture.- 8 Reduction of Sections.- 9 The Space of Sections in the Arithmetic Case and the Section Conjecture in Covers.- Part III On the Passage from Local to Global.- 10 Local Obstructions at a p-adic Place.- 11 Brauer-Manin and Descent Obstructions.- 12 Fragments of Non-abelian Tate–Poitou Duality.- Part IV Analogues of the Section Conjecture.- 13 On the Section Conjecture for Torsors.- 14 Nilpotent Sections.- 15 Sections over Finite Fields.- 16 On the Section Conjecture over Local Fields.- 17 Fields of Cohomological Dimension 1.- 18 Cuspidal Sections and Birational Analogues.

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Book SynopsisThe concept of moduli goes back to B. Riemann, who shows in [68] that the isomorphism class of a Riemann surface of genus 9 ~ 2 depends on 3g - 3 parameters, which he proposes to name "moduli". A precise formulation of global moduli problems in algebraic geometry, the definition of moduli schemes or of algebraic moduli spaces for curves and for certain higher dimensional manifolds have only been given recently (A. Grothendieck, D. Mumford, see [59]), as well as solutions in some cases. It is the aim of this monograph to present methods which allow over a field of characteristic zero to construct certain moduli schemes together with an ample sheaf. Our main source of inspiration is D. Mumford's "Geometric In­ variant Theory". We will recall the necessary tools from his book [59] and prove the "Hilbert-Mumford Criterion" and some modified version for the stability of points under group actions. As in [78], a careful study of positivity proper­ ties of direct image sheaves allows to use this criterion to construct moduli as quasi-projective schemes for canonically polarized manifolds and for polarized manifolds with a semi-ample canonical sheaf.Table of ContentsLeitfaden.- Classification Theory and Moduli Problems.- Notations and Conventions.- 1 Moduli Problems and Hilbert Schemes.- 1.1 Moduli Functors and Moduli Schemes.- 1.2 Moduli of Manifolds: The Main Results.- 1.3 Properties of Moduli Functors.- 1.4 Moduli Functors for ?-Gorenstein Schemes.- 1.5 A. Grothendieck’s Construction of Hilbert Schemes.- 1.6 Hilbert Schemes of Canonically Polarized Schemes.- 1.7 Hilbert Schemes of Polarized Schemes.- 2 Weakly Positive Sheaves and Vanishing Theorems.- 2.1 Coverings.- 2.2 Numerically Effective Sheaves.- 2.3 Weakly Positive Sheaves.- 2.4 Vanishing Theorems and Base Change.- 2.5 Examples of Weakly Positive Sheaves.- 3 D. Mumford’s Geometric Invariant Theory.- 3.1 Group Actions and Quotients.- 3.2 Linearizations.- 3.3 Stable Points.- 3.4 Properties of Stable Points.- 3.5 Quotients, without Stability Criteria.- 4 Stability and Ampleness Criteria.- 4.1 Compactifications and the Hilbert-Mumford Criterion.- 4.2 Weak Positivity of Line Bundles and Stability.- 4.3 Weak Positivity of Vector Bundles and Stability.- 4.4 Ampleness Criteria.- 5 Auxiliary Results on Locally Free Sheaves and Divisors.- 5.1 O. Gabber’s Extension Theorem.- 5.2 The Construction of Coverings.- 5.3 Singularities of Divisors.- 5.4 Singularities of Divisors in Flat Families.- 5.5 Vanishing Theorems and Base Change, Revisited.- 6 Weak Positivity of Direct Images of Sheaves.- 6.1 Variation of Hodge Structures.- 6.2 Weakly Semistable Reduction.- 6.3 Applications of the Extension Theorem.- 6.4 Powers of Dualizing Sheaves.- 6.5 Polarizations, Twisted by Powers of Dualizing Sheaves.- 7 Geometric Invariant Theory on Hilbert Schemes.- 7.1 Group Actions on Hilbert Schemes.- 7.2 Geometric Quotients and Moduli Schemes.- 7.3 Methods to Construct Quasi-Projective Moduli Schemes.- 7.4 Conditions for the Existence of Moduli Schemes: Case (CP).- 7.5 Conditions for the Existence of Moduli Schemes: Case (DP).- 7.6 Numerical Equivalence.- 8 Allowing Certain Singularities.- 8.1 Canonical and Log-Terminal Singularities.- 8.2 Singularities of Divisors.- 8.3 Deformations of Canonical and Log-Terminal Singularities.- 8.4 Base Change and Positivity.- 8.5 Moduli of Canonically Polarized Varieties.- 8.6 Moduli of Polarized Varieties.- 8.7 Towards Moduli of Canonically Polarized Schemes.- 9 Moduli as Algebraic Spaces.- 9.1 Algebraic Spaces.- 9.2 Quotients by Equivalence Relations.- 9.3 Quotients in the Category of Algebraic Spaces.- 9.4 Construction of Algebraic Moduli Spaces.- 9.5 Ample Line Bundles on Algebraic Moduli Spaces.- 9.6 Proper Algebraic Moduli Spaces for Curves and Surfaces.- References.- Glossary of Notations.

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Book SynopsisThis book explores the theory and application of locally nilpotent derivations, a subject motivated by questions in affine algebraic geometry and having fundamental connections to areas such as commutative algebra, representation theory, Lie algebras and differential equations. The author provides a unified treatment of the subject, beginning with 16 First Principles on which the theory is based. These are used to establish classical results, such as Rentschler's Theorem for the plane and the Cancellation Theorem for Curves.More recent results, such as Makar-Limanov's theorem for locally nilpotent derivations of polynomial rings, are also discussed. Topics of special interest include progress in classifying additive actions on three-dimensional affine space, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem.A lot of new material is included in this expanded second edition, such as canonical factorization of quotient morphisms, and a more extended treatment of linear actions. The reader will also find a wealth of examples and open problems and an updated resource for future investigations.Table of ContentsIntroduction.- 1 First Principles.- 2 Further Properties of LNDs.- 3 Polynomial Rings.- 4 Dimension Two.- 5 Dimension Three.- 6 Linear Actions of Unipotent Groups.- 7 Non-Finitely Generated Kernels.- 8 Algorithms.- 9 Makar-Limanov and Derksen Invariants.- 10 Slices, Embeddings and Cancellation.- 11 Epilogue.- References.- Index.

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Book SynopsisThis is a textbook meant to be used at the advanced undergraduate or graduate level. It is an introduction to the theory of Lie groups and Lie algebras. The book treats real and p-adic groups in a unified manner. The first chapter outlines preliminary material that is used in the rest of the book. The second chapter is on analytic functions and is of an elementary nature; this material is included to cater to students who may not be familiar with p-adic fields. The third chapter introduces analytic manifolds and contains standard material; the only notable feature being that it covers both real and p-adic analytic manifolds. Chapters 4 and 5 are on Lie groups. All the standard results on Lie groups are proved here. Some of the proofs are different from those in the earlier literature. The last two chapters are on Lie algebras and cover their structure theory as found in the first of the Bourbaki volumes on the subject. Some proofs here are new.

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Book SynopsisQuesto breve libro propone con uno spirito via via d’immagine storiografica e di dettaglio matematico, la nascita e l’evoluzione del concetto di curvatura: le sue origini ancestrali nella meccanica, nell’astronomia, nella geodesia, e infine, chiaramente nella geometria. Gli aspetti tecnici, a volte estremamente semplici, altre volte complessi, sono sempre accompagnati da spiegazioni che si sperano esaurienti.È ben noto che su entrambi i versanti culturali proposti nel libro, molto si è scritto e ad altissimo livello; qui, c’è un tentativo di sintesi, della storiografia e della matematica sul tema della curvatura. Il racconto del filo che intercorre tra Huygens, Gauss, Riemann, Christoffel, Ricci Curbastro, Levi-Civita e infine Einstein, è stato sicuramente già ben proposto sul versante puramente storico o in quello prettamente matematico: è una speranza che la narrazione qui presentata, con questi punti di vista intrecciati, sia infine soddisfacente. Il tentativo andava fatto. L’augurio forte è che gli argomenti narrati risultino coinvolgenti per il lettore, spingendolo ad esplorare autonomamente altri aspetti magari nascosti nelle pieghe della nozione di curvatura e del mondo che ci vive attorno. Il volume muove inizialmente dal racconto di qualche frammento di cosmologia antica e medioevale. Tutto ciò è solo apparentemente estraneo al corpo vivo della materia: ritroveremo per esempio che la concezione cosmologica di Dante, riassunta qui matematicamente, propose un universo come un’ipersfera 3-dimensionale che, quasi incidentalmente, risulterà proprio il modello cosmologico offerto da Einstein nel 1917 per il suo universo chiuso e statico. Ed è proprio la curvatura che domina quella scena, oggetto matematico protagonista della teoria della relatività generale einsteniana. I personaggi prima elencati vengono comunque narrati anche nelle loro salienti vicende umane, a volte altamente drammatiche, come accadde per esempio per Riemann e Tullio Levi-Civita. In un certo senso, la storia della curvatura accompagna la storia dell’umanità.Benché inizialmente sia stato generato da un disegno didattico, il volume è indirizzato ad un pubblico non necessariamente studentesco, con una cultura scientifica di base.Table of Contents1 Tracce di cosmologia.- 2 Prima di Gauss.- 3 Gauss.- 4 Riemann.- 5 Christoffel.- 6 Ricci Curbastro.- 7 Levi-Civita.- 8 Tracce di geometria differenziale.- 9 Einstein.

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Book SynopsisThis thesis deals with specific features of the theory of holomorphic dynamics in dimension 2 and then sets out to study analogous questions in higher dimensions, e.g. dealing with normal forms for rigid germs, and examples of Kato 3-folds.The local dynamics of holomorphic maps around critical points is still not completely understood, in dimension 2 or higher, due to the richness of the geometry of the critical set for all iterates.In dimension 2, the study of the dynamics induced on a suitable functional space (the valuative tree) allows a classification of such maps up to birational conjugacy, reducing the problem to the special class of rigid germs, where the geometry of the critical set is simple. In some cases, from such dynamical data one can construct special compact complex surfaces, called Kato surfaces, related to some conjectures in complex geometry.Table of ContentsIntroduction.-1.Background.- 2.Dynamics in 2D.- 3.Rigid germs in higher dimension.- 4 Construction of non-Kahler 3-folds.- References.- Index.

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Book SynopsisSeveral generations of students of algebraic geometry have learned the subject from David Mumford's fabled "Red Book" containing notes of his lectures at Harvard University. Their genesis and evolution are described in the preface as:Initially notes to the course were mimeographed and bound and sold by the Harvard math department with a red cover. These old notes were picked up by Springer and are now sold as the "Red book of Varieties and Schemes". However, every time I taught the course, the content changed and grew. I had aimed to eventually publish more polished notes in three volumes...This book contains what Mumford had then intended to be Volume II. It covers the material in the "Red Book" in more depth with several more topics added. The notes have been brought to the present form in collaboration with T. Oda.

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Book SynopsisThis volume contains the proceedings of the international colloquium organized by the Tata Institute of Fundamental Research in January 2016, one of a series of colloquia going back to 1956.The talks at the colloquium covered a wide spectrum of mathematics, ranging over algebraic geometry, topology, algebraic $K$-theory and number theory. Algebraic theory, $\mathbb{A}^1$-homotopy theory and topological $K$-theory formed important sub-streams in this colloquium. Several branches of $K$-theory, like algebraic cycles, triangulated categories of motives, motivic cohomology, motivic homotopy theory, Chow groups of varieties, Euler class theory, equivariant $K$-theory as well as classical $K$-theory have developed considerably in recent years, giving rise to newer directions to the subject as well as proving results of ``classical'' interest. The colloquium brought together experts in these various branches and their talks covered this wide spectrum, highlighting the interconnections and giving a better perspective of the whole subject area.This volume contains refereed articles by leading experts in these fields and includes original results as well as expository materials in these areas.

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Book SynopsisThis book deals with the classical theory of Nevanlinna on the value distribution of meromorphic functions of one complex variable, based on minimum prerequisites for complex manifolds. The theory was extended to several variables by S. Kobayashi, T. Ochiai, J. Carleson, and P. Griffiths in the early 1970s. K. Kodaira took up this subject in his course at The University of Tokyo in 1973 and gave an introductory account of this development in the context of his final paper, contained in this book. The first three chapters are devoted to holomorphic mappings from C to complex manifolds. In the fourth chapter, holomorphic mappings between higher dimensional manifolds are covered. The book is a valuable treatise on the Nevanlinna theory, of special interests to those who want to understand Kodaira's unique approach to basic questions on complex manifolds.Table of ContentsPreface1. Nevanlinna Theory of One Variable (1)1.1 metrics of compact Rimann surfaces1.2 integral formula1.3 holomorphic maps over compact Riemann surfaces whose genus are greater than 21.4 holomorphic maps over Riemann sphreres1.5 Defect relation2. Schwarz--Kobayashi's Lemma2.1 Schwarz--Kobayashi's Lemma2.2 holomorphic maps over algebraic varieties (general type)2.3 hyperbolic measures3. Nevanlinna Theory of One Variable (2)3.1 holomorphic maps over Riemann shpres3.2 the first main theorem3.3 the second main theorem4. Nevanlinna Theory of Several Variables4.1 Biebelbach's example4.2 the first main theorem4.3 the second main theorem4.4 defect relation4.5 applicationsReferences

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Book Synopsis0 Preliminaries.- 1 Enriques surfaces: generalities.- 2 Linear Systems on Enriques Surfaces.- 3 Projective Models of Enriques Surfaces.- 4 Genus One Fibrations.- 5 Moduli Spaces.- Appendix A: Automorphic Forms and Moduli Spaces by S. Kondo.

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Book SynopsisOverview of Work on Sigma Function from Historical Viewpoint.- Curves in Weierstrass Canonical Form (W-curves).- Theory of Sigma Function.- Application of the Sigma Function Theory to Integrable Systems.

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a huge range and FREE tracked UK delivery on ALL orders.

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Book SynopsisThis is the first volume of a two-volume book that offers an in-depth, and essentially self-contained, treatment of the arithmetic theory of algebraic groups. It is accessible to graduate students and researchers in number theory, algebraic geometry, and related areas.Trade Review'The original English version of the book 'Algebraic Groups and Number Theory' by Platonov and Rapinchuk was a go to reference for graduate students and senior researchers alike working in areas of arithmetic and algebraic groups, discrete subgroups of Lie groups, and connections with number theory. The second edition, which will be split into two volumes, and also co-authored with I. Rapinchuk, is a welcome and timely update to the original. The first volume of the second edition, consists of an update to chapters 1-5 of the original with an additional section 4.9 to include new material on the structure of extensions of arithmetic groups. There is no doubt in my mind that this first volume of the second edition will again take on the role of a go to text for those working in an area of huge ongoing interest and importance, and be at the forefront training new generations of mathematicians working in the areas of arithmetic and algebraic groups, discrete subgroups of Lie groups, and connections with number theory.' Alan Reid, Rice University'The arithmetic theory of algebraic groups is a beautiful area of mathematics: a crossroad of number theory, groups, geometry, representation theory, and more. Not surprisingly it attracted some of the greatest mathematicians of the last few generations. The first edition of the book 'Algebraic Groups and Number Theory' by Vladimir Platonov and Andrei Rapinchuk which came out in the early 90s has quickly become the standard reference of the field. It presents in a clear way several deep topics. The book was one of the reasons the area attracted more researchers and expanded to new directions. This made an updated version much needed. The original authors and Igor Rapinchuk should be thanked by the mathematical community for carrying out this monumental job.' Alex Lubotzky, Hebrew University of JerusalemTable of Contents1. Algebraic number theory; 2. Algebraic groups; 3. Algebraic groups over locally compact fields; 4. Arithmetic groups and reduction theory; 5. Adeles; Bibliography; Index.

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Book SynopsisAlgebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. This short and readable introduction to algebraic geometry will be ideal for all undergraduate mathematicians coming to the subject for the first time. With the minimum of prerequisites, Dr Reid introduces the reader to the basic concepts of algebraic geometry including: plane conics, cubics and the group law, affine and projective varieties, and non-singularity and dimension. He is at pains to stress the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book arises from an undergraduate course given at the University of Warwick and contains numerous examples and exercises illustrating the theory.Trade Review"Before Reid's volume there was hardly anything to recommend at the undergraduate level...Reid's book is fun; it is filled with examples, applications, asides, gossip...What it does, it does well, and there is nothing comparable." Choice"...at a level advanced undergraduates will understand and appreciate." Mathematics Magazine"...the author leads the student on a lively, interesting, down-to-earth tour of the fundamental algebraic geometry...with some welcome, provocative comments..." American Mathematical MonthlyTable of Contents1. Playing with plane curves; 2. The category of affine varieties; 3. Applications; Index.

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