Download or read book General Existence Theorems in Moduli Theory written by Jack Kingsbury Hall and published by Stanford University. This book was released on 2011 with total page 151 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we prove that there is an algebraic stack parameterizing all curves. The curves that appear in this algebraic stack are allowed to be arbitrarily singular, non-reduced, disconnected, and reducible. We also prove the boundedness of the open substack parameterizing reduced and connected curves with fixed arithmetic genus g and at most e irreducible components. We also show that for essentially any algebraic stack, there is an algebraic stack, the Hilbert stack, parameterizing quasi-finite maps to the stack. The technical heart of this result is a generalization of formal GAGA to a non-separated morphism of algebraic stacks, something that was previously unknown for a morphism of schemes. We also employ derived algebraic geometry, in an essential way, to prove the algebraicity of the Hilbert stack. The Hilbert stack, for algebraic spaces, was claimed to exist by M. Artin (1974), but was left unproved due to a lack of foundational results for non-separated algebraic spaces. Finally, we generalize the fundamental GAGA results of J. P. Serre (1956) in three ways---to the non-separated setting, to stacks, and to families. As an application of these results, we show that analytic compactifications of the moduli stack of smooth curves possessing modular interpretations are algebraizable.
Download or read book General Existence Theorems in Moduli Theory written by Jack Kingsbury Hall and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we prove that there is an algebraic stack parameterizing all curves. The curves that appear in this algebraic stack are allowed to be arbitrarily singular, non-reduced, disconnected, and reducible. We also prove the boundedness of the open substack parameterizing reduced and connected curves with fixed arithmetic genus g and at most e irreducible components. We also show that for essentially any algebraic stack, there is an algebraic stack, the Hilbert stack, parameterizing quasi-finite maps to the stack. The technical heart of this result is a generalization of formal GAGA to a non-separated morphism of algebraic stacks, something that was previously unknown for a morphism of schemes. We also employ derived algebraic geometry, in an essential way, to prove the algebraicity of the Hilbert stack. The Hilbert stack, for algebraic spaces, was claimed to exist by M. Artin (1974), but was left unproved due to a lack of foundational results for non-separated algebraic spaces. Finally, we generalize the fundamental GAGA results of J.P. Serre (1956) in three ways--to the non-separated setting, to stacks, and to families. As an application of these results, we show that analytic compactifications of the moduli stack of smooth curves possessing modular interpretations are algebraizable.
Download or read book The Geometry of Moduli Spaces of Sheaves written by Daniel Huybrechts and published by Cambridge University Press. This book was released on 2010-05-27 with total page 345 pages. Available in PDF, EPUB and Kindle. Book excerpt: This edition has been updated to reflect recent advances in the theory of semistable coherent sheaves and their moduli spaces. The authors review changes in the field and point the reader towards further literature. An ideal text for graduate students or mathematicians with a background in algebraic geometry.
Download or read book The Moduli Space of Curves written by Robert H. Dijkgraaf and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 570 pages. Available in PDF, EPUB and Kindle. Book excerpt: The moduli space Mg of curves of fixed genus g – that is, the algebraic variety that parametrizes all curves of genus g – is one of the most intriguing objects of study in algebraic geometry these days. Its appeal results not only from its beautiful mathematical structure but also from recent developments in theoretical physics, in particular in conformal field theory.
Download or read book Moduli Theory and Classification Theory of Algebraic Varieties written by H. Popp and published by Springer. This book was released on 2006-11-15 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Local Moduli and Singularities written by Olav Arnfinn Laudal and published by Springer. This book was released on 2006-11-15 with total page 123 pages. Available in PDF, EPUB and Kindle. Book excerpt: This research monograph sets out to study the notion of a local moduli suite of algebraic objects like e.g. schemes, singularities or Lie algebras and provides a framework for this. The basic idea is to work with the action of the kernel of the Kodaira-Spencer map, on the base space of a versal family. The main results are the existence, in a general context, of a local moduli suite in the category of algebraic spaces, and the proof that, generically, this moduli suite is the quotient of a canonical filtration of the base space of the versal family by the action of the Kodaira-Spencer kernel. Applied to the special case of quasihomogenous hypersurfaces, these ideas provide the framework for the proof of the existence of a coarse moduli scheme for plane curve singularities with fixed semigroup and minimal Tjurina number . An example shows that for arbitrary the corresponding moduli space is not, in general, a scheme. The book addresses mathematicians working on problems of moduli, in algebraic or in complex analytic geometry. It assumes a working knowledge of deformation theory.
Download or read book Moduli in Modern Mapping Theory written by Olli Martio and published by Springer Science & Business Media. This book was released on 2008-11-09 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on recent research papers, this book presents a modern account of mapping theory with emphasis on quasiconformal mapping and its generalizations. It contains an extensive bibliography.
Download or read book Moduli of Curves written by Joe Harris and published by Springer Science & Business Media. This book was released on 2006-04-06 with total page 381 pages. Available in PDF, EPUB and Kindle. Book excerpt: A guide to a rich and fascinating subject: algebraic curves and how they vary in families. Providing a broad but compact overview of the field, this book is accessible to readers with a modest background in algebraic geometry. It develops many techniques, including Hilbert schemes, deformation theory, stable reduction, intersection theory, and geometric invariant theory, with the focus on examples and applications arising in the study of moduli of curves. From such foundations, the book goes on to show how moduli spaces of curves are constructed, illustrates typical applications with the proofs of the Brill-Noether and Gieseker-Petri theorems via limit linear series, and surveys the most important results about their geometry ranging from irreducibility and complete subvarieties to ample divisors and Kodaira dimension. With over 180 exercises and 70 figures, the book also provides a concise introduction to the main results and open problems about important topics which are not covered in detail.
Download or read book An Introduction to Invariants and Moduli written by Shigeru Mukai and published by Cambridge University Press. This book was released on 2003-09-08 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sample Text
Download or read book Moduli Spaces of Riemann Surfaces written by Benson Farb and published by American Mathematical Soc.. This book was released on 2013-08-16 with total page 371 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mapping class groups and moduli spaces of Riemann surfaces were the topics of the Graduate Summer School at the 2011 IAS/Park City Mathematics Institute. This book presents the nine different lecture series comprising the summer school, covering a selection of topics of current interest. The introductory courses treat mapping class groups and Teichmüller theory. The more advanced courses cover intersection theory on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable cohomology of mapping class groups, the structure of Torelli groups, and arithmetic mapping class groups. The courses consist of a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The book should be a valuable resource for graduate students and researchers interested in the topology, geometry and dynamics of moduli spaces of Riemann surfaces and related topics. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.
Download or read book Existence Theorems for Minimal Surfaces of Non Zero Genus Spanning a Contour written by Friedrich Tomi and published by American Mathematical Soc.. This book was released on 1988 with total page 93 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present a modern approach to the classical problem of Plateau based purely on differential geometric concepts. We not only reprove the classical results of Douglas but also develop a new geometric criterion on a given finite system of disjoint Jordan curves in three-dimensional Euclidean space which guarantees the existence of a minimal surface of a prescribed genus having these curves as boundary.
Download or read book Quasi projective Moduli for Polarized Manifolds written by Eckart Viehweg and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: The 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.
Download or read book Relative Moduli Spaces of Semi stable Sheaves on Families of Curves written by Jens Thomas Alexander Lang and published by Herbert Utz Verlag. This book was released on 2001 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book A Panoramic View of Riemannian Geometry written by Marcel Berger and published by Springer Science & Business Media. This book was released on 2007-06-29 with total page 852 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field. From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." --MATHEMATICAL REVIEWS
Download or read book Notices of the American Mathematical Society written by American Mathematical Society and published by . This book was released on 1988 with total page 810 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Metrics Connections and Gluing Theorems written by Clifford Taubes and published by American Mathematical Soc.. This book was released on 1996 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, the author's goal is to provide an introduction to some of the analytic underpinnings for the geometry of anti-self duality in 4-dimensions. Anti-self duality is rather special to 4-dimensions and the imposition of this condition on curvatures of connections on vector bundles and on curvatures of Riemannian metrics has resulted in some spectacular mathematics. The book reviews some basic geometry, but is is assumed that the reader has a general background in differential geometry (as would be obtained by reading a standard text on the subject). Some of the fundamental references include Atiyah, Hitchin and Singer, Freed and Uhlenbeck, Donaldson and Kronheimer, and Kronheimer and Mrowka. The last chapter contains open problems and conjectures.
Download or read book Lectures on K3 Surfaces written by Daniel Huybrechts and published by Cambridge University Press. This book was released on 2016-09-26 with total page 499 pages. Available in PDF, EPUB and Kindle. Book excerpt: K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.
