Deformations of algebraic schemes via reedypalamodov. Deformations of algebraic schemes via reedypalamodov cofibrant. In this thesis, we study deformations of compact holomorphic poisson manifolds and algebraic poisson schemes in the framework of kodairaspencers analytic deformation theory and grothendiecks algebraic deformation theory. Sep 01, 2015 read invariant deformation theory of affine schemes with reductive group action, journal of pure and applied algebra on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. The goal of this paper is to study deformations of x over a differential graded local artin \mathbbkalgebra by using local tatequillen. Deformations of algebraic varieties with gm action numdam. I am asking this question in an attempt to clarify my vague understanding, especially in regard to the last bullet point below. We focus on deformations of schemes with an action by a nite group g over an algebraically closed eld of characteristic p0. Jacobibernoulli cohomology and deformations of schemes and. Matsusaka, algebraic deformations of polarized varieties, nagoya math.
This book brings together both the classical and current aspects of deformation theory. This book aims at giving an account with complete proofs of the\ud results and techniques which are needed to understand the local\ud deformation theory of algebraic schemes over an algebraically\ud closed field, thus providing the tools needed for example in the\ud local study of hilbert schemes and moduli problems. Section 5 of fundamental algebraic geometry has a construction of the hilbert and quot schemes. In particular, this provides an evidence toward a conjecture of debarre. In this case the transverse deformations are resolutions of deformations of double covers of projective space but not double covers of a blowup of projective space. Deformations of algebraic schemes edoardo sernesi this account of deformation theory in classical algebraic geometry over an algebraically closed field presents for the first time some results previously scattered in the literature, with proofs that are relatively little known, yet relevant to algebraic geometers. The following counterexample has been kindly provided by m.
For full access to this pdf, sign in to an existing account, or purchase an annual subscription. Request pdf on jan 1, 2006, edoardo sernesi and others published deformations of algebraic schemes find, read and cite all the research you need on researchgate. Read invariant deformation theory of affine schemes with reductive group action, journal of pure and applied algebra on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. We study the birational geometry of deformations of hilbert schemes of points on p2. The tame fundamental group of a formal neigh borhood of a divisor with normal crossings on a scheme, springerverlag, lecture notes in math 208, berlin 1971. Spencer, after deformation techniques had received a great deal of more tentative application in the italian school of algebraic geometry. Deformations of lie algebras of vector fields arising from.
A series of comprehensive studies in mathematics, vol 334. To that end, it introduces the notion of inf scheme, which is an infinitesimal deformation of a scheme and studies indcoherent sheaves on such. The goal of this paper is to study deformations of x over a differential graded local artin kalgebra by using local tatequillen resolutions, i. On the homological algebra of relative symplectic geometry pomerleano, daniel, 2019. Contents terminology and notation introduction infinitesimal deformations 9 1.
Deformations of algebraic schemes edoardo sernesi 2 preface in some sense deformation theory is as old as algebraic geometry itself. Seshadri and b cohomology of certain moduli spaces of vector bundles proc. Deformations of minimal cohomology classes on abelian. Deformations of algebraic schemes via reedypalamodov cofibrant resolutions. There isnt much deformation theory in this chapter currently. We study the deformation theory of algebraic stacks. The above goal is achieved by describing the dglie algebra. The 2category of deformations using ext groups of the cotangent complex is described. As an application, we prove that if a jacobian j deforms together with a minimal cohomology class out the jacobian locus, then j is hyperelliptic. Deformations of algebraic schemes via reedypalamodov cofibrant resolutions authors.
Deformations of compact holomorphic poisson manifolds and algebraic poisson schemes. We construct the deformation functor associated with a pair of morphisms of differential graded lie algebras, and use it to study infinitesimal deformations of holomorphic maps of compact complex manifolds. Additional gift options are available when buying one ebook at a time. Later browse through fundamental algebraic geometry. Deformations of compact holomorphic poisson manifolds and.
Deformations of algebraic schemes edoardo sernesi springer. In the interest of readability, some technically complicated proofs have been omitted when a suitable reference was. Nevertheless, a correct understanding of what deforming means leads into the technically most dif. It does a great job complementing hartshornes treatment of schemes, above all because. Studying deformation theory of schemes stack exchange. Ample subvarieties of algebraic varieties download link. These are expanded lecture notes for the authors minicourse at msri in june 2012.
Pdf nonsingular deformations of a determinantal scheme. Marco manetti, francesco meazzini submitted on 19 oct 2018 v1, last revised 4 sep 2019 this version, v3. In one sense, deformation theory is as old as algebraic geometry itself. In the appendix, we study deformations of poisson vector bundles. Deformations of algebras in noncommutative algebraic geometry travis schedler abstract.
Unobstructedness of deformations of weak fano manifolds. It has found numerous applications in other parts of mathematics, most prominently in representation theory. Deformations of algebraic schemes edoardo sernesi 2 preface in some sense deformation theory is as old as algebraic. We show that the infinitesimal deformations of brillnoether loci w d attached to a smooth nonhyperelliptic curve c are in onetone correspondence with the deformations of c. One important class of such deformation problems would be the deformations of a pair x, f, where x is a smooth algebraic variety and f is a coherent sheaf over x, which was studied in detail in. This gives a method to compute the hodge numbers of these calabiyau manifolds. Let x be a noetherian separated and finite dimensional scheme over a field k of characteristic zero. Finally, we give some of pinkhams results on deformations of cones over rational curves cf.
The minimal model program for deformations of hilbert. A guide to the literature on algebraic stacks 5 this generalizes standard deformation theory results for morphisms of schemes to representable morphisms of algebraic stacks in terms of the cotangent complex. In algebraic geometry, a degeneration or specialization is the act of taking a limit of a family of varieties. Artin and others have studied deformations of singularities. Altogether, these additional topics amount to nearly half the book, and they are included here both to make the book more comprehensive and. Updated to reflect current research, algebraic number theory and fermats last theorem, fourth edition introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematicsthe quest for a proof of fermats last theorem. Chunyi li, xiaolei zhao submitted on 6 dec 20 abstract. We study the birational geometry of deformations of hilbert schemes of points on the projective plane p2.
Wedhorn algebraic geometry i, schemes with examples and exercises. We introduce a notion of jacobibernoulli cohomology associated to a semisimplicial lie algebra sela. Ample subvarieties of algebraic varieties free book at ebooks directory. Li, chunyi and zhao, xiaolei 2018 the minimal model program for deformations of hilbert schemes of points on the projective plane. The minimal model program for deformations of hilbert schemes. Nonsingular deformations of a determinantal scheme. Deformations of algebraic schemes pdf free download. Download it once and read it on your kindle device, pc, phones or tablets. As a beginning differential geometer, ive been trying to learn about deformation theory.
The book is recommended for advanced graduate students and researchers in algebraic geometry who want to learn deformation theory. Pdf topics in algebraic deformation theory researchgate. Deformations of algebraic schemes edoardo sernesi download. The goal of this paper is to study deformations of x over a differential graded local artin k algebra by using local tatequillen resolutions, i. All my papers in this field have been published by springerverlag in two volumes, a selected papers on the classification of varieties and moduli spaces, and b selected papers ii, on algebraic geometry including correspondence with grothendieck.
Pdf algebraic number theory and fermat s last theorem. Bernoulli numbers and deformations of schemes and maps. Most of the results in this area thus far concern actions on smooth curves. These notes are based on a series of lectures given at the tata institute in januaryfebruary, 1973. Other than kodairas book, ive found virtually no references from the point of view of differential geometry. Issuu is a digital publishing platform that makes it simple to publish magazines, catalogs, newspapers, books, and more online.
The mmp for deformations of hilbert schemes of points on the projective plane. Deformation theory is the study of the behaviour of a family of algebraic geometric objects, such as schemes or coherent sheaves, near a given element. We show that moduli of bridgelandstable objects are smooth, irreducible, projective varieties, which are birationally equivalent to these deformations. This volume develops deformation theory, lie theory and the theory of algebroids in the context of derived algebraic geometry.
A number of significant changes appear after world war ii. Then we show that an algebraic stack has a versal deformation under certain conditions. One motivation to look at deformations comes from physics, see for example kontsevichs famous paper on deformation quantization of poisson manifolds. The presentation is mostly selfcontained, assuming only basic knowledge of commutative algebra, homological algebra and category theory. Apr 20, 2007 in one sense, deformation theory is as old as algebraic geometry itself.
The paper investigates the deformations of a determinantal scheme arising from a deformation of the defining matrix. The most salient deformation theory in mathematics has been that of complex manifolds and algebraic varieties. This article is a continuation of previous work, which has the same title. Download now updated to reflect current research, algebraic number theory and fermats last theorem, fourth edition introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematicsthe quest for a proof of fermats last theorem. Algebraic and geometric isomonodromic deformations doran, charles f. These results cannot be viewed as consequences of illusies general theory as the cotangent complex of a representable mor.
This was put on a firm basis by foundational work of kunihiko kodaira and donald c. We identify firstorder deformations and obstructions. Moreover, wall crossing in the space of bridgelandstability conditions induces the minimal model program for these deformations. Lectures on deformations of singularities download link. We study deformations of nonsingular poisson varieties and poisson invertible sheaves, which extend the flat deformation theory of nonsingular varieties and invertible sheaves. For deformation theory in algebraic geometry, try taking a look at moduli of curves by harrismorrison, deformations of algebraic schemes by sernesi, or these notes of hartshorne. The study of small and local deformations of algebraic varieties originates in the classical work of kodaira and spencer and its. In mathematics, deformation theory is the study of infinitesimal conditions associated with. It has now been four decades since david mumford wrote that algebraic geometry seems to have acquired the. It includes applications to the construction and properties of severi varieties of families of plane nodal curves, space curves, deformations of quotient singularities, hilbert schemes of points, local picard functors, etc.
Infinitesimal deformations of double covers of smooth. Pdf we give a selective survey of topics in algebraic deformation theory ranging from its inception to current times. Fano threefolds with anticanonical du val k 3 surfaces. Indagationes mathematicae vol 31, issue 1, pages 1184. Easily share your publications and get them in front of issuus. Deformations of algebraic schemes edoardo sernesi 2 preface in some sense deformation theory is as old as.
Artins versal deformations and algebraic stacks inventions mathematicae, vol. Derived algebraic geometry is a farreaching generalization of algebraic geometry. Versal property local deformation space miniversal deformation space i came across these words while studying these papers a desingularization of moduli varities for vector bundles on curves, i. Jacobibernoulli cohomology and deformations of schemes and maps article in central european journal of mathematics 104 july 2008 with 7 reads how we measure reads.
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