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Arxiv:1908.07480V3 [Math.AG]
TORSORS ON LOOP GROUPS AND THE HITCHIN FIBRATION ALEXIS BOUTHIER AND KĘSTUTIS ČESNAVIČIUS Abstract. In his proof of the fundamental lemma, Ngô established the product formula for the Hitchin fibration over the anisotropic locus. One expects this formula over the larger generically regular semisimple locus, and we confirm this by deducing the relevant vanishing statement for torsors over Rpptqq from a general formula for PicpRpptqqq. In the build up to the product formula, we present general algebraization, approximation, and invariance under Henselian pairs results for torsors, give short new proofs for the Elkik approximation theorem and the Chevalley isomorphism g G – t{W , and improve results on the geometry of the Chevalley morphism g Ñ g G. Résumé. Dans sa preuve du lemme fondamental, Ngô établit une formule du produit au-dessus du lieu anisotrope. On s’attend à ce qu’une telle formule s’étende au-dessus de l’ouvert génériquement régulier semisimple. Nous établissons une telle formule en la déduisant d’un résultat d’annulation de torseurs sous des groupes de lacets à partir d’une formule générale pour PicpRpptqqq. Dans le cours de la preuve, nous montrons des résultats généraux d’algébrisation, d’approximation et d’invariance hensélienne pour des torseurs ; nous donnons de nouvelles preuves plus concises du théorème d’algébrisation d’Elkik et de l’isomorphisme de Chevalley g G – t{W et améliorons les énoncés sur la géométrie du morphisme de Chevalley g Ñ g G. 1. Introduction ................................................... ...................... 2 Acknowledgements ................................... ................................ 4 2. Approximation and algebraization ................................................. 5 2.1. Invariance of torsors under Henselian pairs and algebraization ................... -
NOTES on CARTIER and WEIL DIVISORS Recall: Definition 0.1. A
NOTES ON CARTIER AND WEIL DIVISORS AKHIL MATHEW Abstract. These are notes on divisors from Ravi Vakil's book [2] on scheme theory that I prepared for the Foundations of Algebraic Geometry seminar at Harvard. Most of it is a rewrite of chapter 15 in Vakil's book, and the originality of these notes lies in the mistakes. I learned some of this from [1] though. Recall: Definition 0.1. A line bundle on a ringed space X (e.g. a scheme) is a locally free sheaf of rank one. The group of isomorphism classes of line bundles is called the Picard group and is denoted Pic(X). Here is a standard source of line bundles. 1. The twisting sheaf 1.1. Twisting in general. Let R be a graded ring, R = R0 ⊕ R1 ⊕ ::: . We have discussed the construction of the scheme ProjR. Let us now briefly explain the following additional construction (which will be covered in more detail tomorrow). L Let M = Mn be a graded R-module. Definition 1.1. We define the sheaf Mf on ProjR as follows. On the basic open set D(f) = SpecR(f) ⊂ ProjR, we consider the sheaf associated to the R(f)-module M(f). It can be checked easily that these sheaves glue on D(f) \ D(g) = D(fg) and become a quasi-coherent sheaf Mf on ProjR. Clearly, the association M ! Mf is a functor from graded R-modules to quasi- coherent sheaves on ProjR. (For R reasonable, it is in fact essentially an equiva- lence, though we shall not need this.) We now set a bit of notation. -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Cox Rings and Partial Amplitude Permalink https://escholarship.org/uc/item/7bs989g2 Author Brown, Morgan Veljko Publication Date 2012 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Cox Rings and Partial Amplitude by Morgan Veljko Brown A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, BERKELEY Committee in charge: Professor David Eisenbud, Chair Professor Martin Olsson Professor Alistair Sinclair Spring 2012 Cox Rings and Partial Amplitude Copyright 2012 by Morgan Veljko Brown 1 Abstract Cox Rings and Partial Amplitude by Morgan Veljko Brown Doctor of Philosophy in Mathematics University of California, BERKELEY Professor David Eisenbud, Chair In algebraic geometry, we often study algebraic varieties by looking at their codimension one subvarieties, or divisors. In this thesis we explore the relationship between the global geometry of a variety X over C and the algebraic, geometric, and cohomological properties of divisors on X. Chapter 1 provides background for the results proved later in this thesis. There we give an introduction to divisors and their role in modern birational geometry, culminating in a brief overview of the minimal model program. In chapter 2 we explore criteria for Totaro's notion of q-amplitude. A line bundle L on X is q-ample if for every coherent sheaf F on X, there exists an integer m0 such that m ≥ m0 implies Hi(X; F ⊗ O(mL)) = 0 for i > q. -
Introduction to Algebraic Geometry
Introduction to Algebraic Geometry Jilong Tong December 6, 2012 2 Contents 1 Algebraic sets and morphisms 11 1.1 Affine algebraic sets . 11 1.1.1 Some definitions . 11 1.1.2 Hilbert's Nullstellensatz . 12 1.1.3 Zariski topology on an affine algebraic set . 14 1.1.4 Coordinate ring of an affine algebraic set . 16 1.2 Projective algebraic sets . 19 1.2.1 Definitions . 19 1.2.2 Homogeneous Nullstellensatz . 21 1.2.3 Homogeneous coordinate ring . 22 1.2.4 Exercise: plane curves . 22 1.3 Morphisms of algebraic sets . 24 1.3.1 Affine case . 24 1.3.2 Quasi-projective case . 26 2 The Language of schemes 29 2.1 Sheaves and locally ringed spaces . 29 2.1.1 Sheaves on a topological spaces . 29 2.1.2 Ringed space . 34 2.2 Schemes . 36 2.2.1 Definition of schemes . 36 2.2.2 Morphisms of schemes . 40 2.2.3 Projective schemes . 43 2.3 First properties of schemes and morphisms of schemes . 49 2.3.1 Topological properties . 49 2.3.2 Noetherian schemes . 50 2.3.3 Reduced and integral schemes . 51 2.3.4 Finiteness conditions . 53 2.4 Dimension . 54 2.4.1 Dimension of a topological space . 54 2.4.2 Dimension of schemes and rings . 55 2.4.3 The noetherian case . 57 2.4.4 Dimension of schemes over a field . 61 2.5 Fiber products and base change . 62 2.5.1 Sum of schemes . 62 2.5.2 Fiber products of schemes . -
Arxiv:1708.05877V4 [Math.AC]
NORMAL HYPERPLANE SECTIONS OF NORMAL SCHEMES IN MIXED CHARACTERISTIC JUN HORIUCHI AND KAZUMA SHIMOMOTO Abstract. The aim of this article is to prove that, under certain conditions, an affine flat normal scheme that is of finite type over a local Dedekind scheme in mixed characteristic admits infinitely many normal effective Cartier divisors. For the proof of this result, we prove the Bertini theorem for normal schemes of some type. We apply the main result to prove a result on the restriction map of divisor class groups of Grothendieck-Lefschetz type in mixed characteristic. Dedicated to Prof. Gennady Lyubeznik on the occasion of his 60th birthday. 1. Introduction Let X be a connected Noetherian normal scheme. Then does X have sufficiently many normal Cartier divisors? The existence of such a divisor when X is a normal projective variety over an algebraically closed field is already known. This fact follows from the classical Bertini theorem due to Seidenberg. In birational geometry, it is often necessary to compare the singularities of X with the singularities of a divisor D X, which is ⊂ known as adjunction (see [12] for this topic). In this article, we prove some results related to this problem in the mixed characteristic case. The main result is formulated as follows (see Corollary 5.4 and Remark 5.5). Theorem 1.1. Let X be a normal connected affine scheme such that there is a surjective flat morphism of finite type X Spec A, where A is an unramified discrete valuation ring → of mixed characteristic p > 0. Assume that dim X 2, the generic fiber of X Spec A ≥ → is geometrically connected and the residue field of A is infinite. -
Arxiv:1711.06456V4 [Math.AG] 1 Dec 2018
PURITY FOR THE BRAUER GROUP KĘSTUTIS ČESNAVIČIUS Abstract. A purity conjecture due to Grothendieck and Auslander–Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension ě 2. The combination of several works of Gabber settles the conjecture except for some cases that concern p-torsion Brauer classes in mixed characteristic p0,pq. We establish the remaining cases by using the tilting equivalence for perfectoid rings. To reduce to perfectoids, we control the change of the Brauer group of the punctured spectrum of a local ring when passing to a finite flat cover. 1. The purity conjecture of Grothendieck and Auslander–Goldman .............. 1 Acknowledgements ................................... ................................ 3 2. Passage to a finite flat cover ................................................... .... 3 3. Passage to the completion ................................................... ....... 6 4. The p-primary Brauer group in the perfectoid case .............................. 7 5. Passage to perfect or perfectoid towers ........................................... 11 6. Global conclusions ................................................... ............... 13 Appendix A. Fields of dimension ď 1 ................................................ 15 References ................................................... ........................... 16 1. The purity conjecture of Grothendieck and Auslander–Goldman Grothendieck predicted in [Gro68b, §6] that the cohomological Brauer group of a regular scheme X is insensitive to removing a closed subscheme Z Ă X of codimension ě 2. This purity conjecture is known in many cases (as we discuss in detail below), for instance, for cohomology classes of order 2 invertible on X, and its codimension requirement is necessary: the Brauer group of AC does not agree with that of the complement of the coordinate axes (see [DF84, Rem. 3]). In this paper, we finish the remaining cases, that is, we complete the proof of the following theorem. -
Moving Codimension-One Subvarieties Over Finite Fields
Moving codimension-one subvarieties over finite fields Burt Totaro In topology, the normal bundle of a submanifold determines a neighborhood of the submanifold up to isomorphism. In particular, the normal bundle of a codimension-one submanifold is trivial if and only if the submanifold can be moved in a family of disjoint submanifolds. In algebraic geometry, however, there are higher-order obstructions to moving a given subvariety. In this paper, we develop an obstruction theory, in the spirit of homotopy theory, which gives some control over when a codimension-one subvariety moves in a family of disjoint subvarieties. Even if a subvariety does not move in a family, some positive multiple of it may. We find a pattern linking the infinitely many obstructions to moving higher and higher multiples of a given subvariety. As an application, we find the first examples of line bundles L on smooth projective varieties over finite fields which are nef (L has nonnegative degree on every curve) but not semi-ample (no positive power of L is spanned by its global sections). This answers questions by Keel and Mumford. Determining which line bundles are spanned by their global sections, or more generally are semi-ample, is a fundamental issue in algebraic geometry. If a line bundle L is semi-ample, then the powers of L determine a morphism from the given variety onto some projective variety. One of the main problems of the minimal model program, the abundance conjecture, predicts that a variety with nef canonical bundle should have semi-ample canonical bundle [15, Conjecture 3.12]. -
Arxiv:1807.03665V3 [Math.AG]
DEMAILLY’S NOTION OF ALGEBRAIC HYPERBOLICITY: GEOMETRICITY, BOUNDEDNESS, MODULI OF MAPS ARIYAN JAVANPEYKAR AND LJUDMILA KAMENOVA Abstract. Demailly’s conjecture, which is a consequence of the Green–Griffiths–Lang con- jecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly’s conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly’s definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a pro- jective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that Aut(X) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore “weaker” notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green–Griffiths–Lang conjecture on hyperbolic projective varieties. 1. Introduction The aim of this paper is to provide evidence for Demailly’s conjecture which says that a projective algebraically hyperbolic variety over C is Kobayashi hyperbolic. We first define the notion of an algebraically hyperbolic projective scheme over an alge- braically closed field k of characteristic zero which is not assumed to be C, and could be Q, for example. Then we provide indirect evidence for Demailly’s conjecture by showing that algebraically hyperbolic schemes share many common features with Kobayashi hyperbolic complex manifolds. -
Arxiv:1706.04845V2 [Math.AG] 26 Jan 2020 Usos Ewudas Iet Hn .Batfrifrigu Bu [B Comments
RELATIVE SEMI-AMPLENESS IN POSITIVE CHARACTERISTIC PAOLO CASCINI AND HIROMU TANAKA Abstract. Given an invertible sheaf on a fibre space between projective varieties of positive characteristic, we show that fibre- wise semi-ampleness implies relative semi-ampleness. The same statement fails in characteristic zero. Contents 1. Introduction 2 1.1. Description of the proof 3 2. Preliminary results 6 2.1. Notation and conventions 6 2.2. Basic results 8 2.3. Dimension formulas for universally catenary schemes 11 2.4. Relative semi-ampleness 13 2.5. Relative Keel’s theorem 18 2.6. Thickening process 20 2.7. Alteration theorem for quasi-excellent schemes 24 3. (Theorem C)n−1 implies (Theorem A)n 26 4. Numerically trivial case 30 4.1. The case where the total space is normal 30 4.2. Normalisation of the base 31 4.3. The vertical case 32 arXiv:1706.04845v2 [math.AG] 26 Jan 2020 4.4. (Theorem A)n implies (Theoerem B)n 37 4.5. Generalisation to algebraic spaces 43 5. (Theorem A)n and (Theorem B)n imply (Theorem C)n 44 6. Proofofthemaintheorems 49 7. Examples 50 2010 Mathematics Subject Classification. 14C20, 14G17. Key words and phrases. relative semi-ample, positive characteristic. The first author was funded by EPSRC. The second author was funded by EP- SRC and the Grant-in-Aid for Scientific Research (KAKENHI No. 18K13386). We would like to thank Y. Gongyo, Z. Patakfalvi and S. Takagi for many useful dis- cussions. We would also like to thank B. Bhatt for informing us about [BS17]. -
Perfectoid Spaces
MATH 679: PERFECTOID SPACES BHARGAV BHATT∗ Course Description. Perfectoid spaces are a class of spaces in arithmetic geometry introduced in 2012 by Peter Scholze in his PhD thesis [Sch12]. Despite their youth, these spaces have had stunning applications to many different areas of mathematics, including number theory, algebraic geometry, representation theory, and commutative algebra. The key to this success is that perfectoid spaces provide a functorial procedure to translate certain algebro-geometric problems from characteristic 0 (or mixed characteristic) to characteristic p; the latter can often be more accessible thanks to the magic of Frobenius. A major portion of this class will be devoted to setting up the basic theory of perfectoid spaces. En route, we will encounter Huber's approach to nonarchimedean geometry via his language of adic spaces, Faltings' theory of `almost mathematics' conceived in his proof of Fontaine's conjectures in p-adic Hodge theory, and the basic algebraic geometry of perfect schemes in characteristic p. The highlight of this part of the course will be the `almost purity theorem', a cruder version of which forms the cornerstone of Faltings' aforementioned work. The rest of the course will focus on a single application of perfectoid spaces. There are several choices here: we could go either in the arithmetic direction (such as Scholze's work on the weight-monodromy conjecture or some recent progress in p-adic Hodge theory) or in a more algebraic direction (which is the lecturer's inclination). A final choice will be made during the semester depending on audience makeup and interest. Contents 1. -
Arxiv:1506.08738V2 [Math.AG] 7 Aug 2015 Let Upre on Supported Ento 1
VARIANTS OF NORMALITY FOR NOETHERIAN SCHEMES JANOS´ KOLLAR´ Abstract. This note presents a uniform treatment of normality and three of its variants—topological, weak and seminormality—for Noetherian schemes. The key is to define these notions for pairs (Z,X) consisting of a (not nec- essarily reduced) scheme X and a closed, nowhere dense subscheme Z. An advantage of the new definitions is that, unlike the usual absolute ones, they are preserved by completions. This shortens some of the proofs and leads to more general results. Definition 1. Let X be a scheme and Z X a closed, nowhere dense subscheme. A finite modification of X centered at Z is⊂ a finite morphism p : Y X such that −1 → none of the associated primes of Y is contained in ZY := p (Z) and p Y : Y ZY X Z is an isomorphism. (1.1) |Y \Z \ → \ Let j : X Z ֒ X be the natural injection and JZ X the largest subsheaf supported\ on Z→. There is a one-to-one correspondence between⊂ O finite modifications and coherent X -algebras O X /JZ p Y j . (1.2) O ⊂ ∗O ⊂ ∗OX\Z The notion of an integral modification of X centered at Z is defined analogously. Let A j∗ X\Z be the largest subalgebra that is integral over X . Then SpecX A is the⊂ maximalO integral modification, called the relative normalizationO of the pair Z X. We denote it by ⊂ π : (Zrn Xrn) (Z X) orby π : Xrn X. (1.3) ⊂ → ⊂ Z → The relative normalization is the limit of all finite modifications centered at Z. -
Grothendieck Ring of Varieties
Neeraja Sahasrabudhe Grothendieck Ring of Varieties Thesis advisor: J. Sebag Université Bordeaux 1 1 Contents 1. Introduction 3 2. Grothendieck Ring of Varieties 5 2.1. Classical denition 5 2.2. Classical properties 6 2.3. Bittner's denition 9 3. Stable Birational Geometry 14 4. Application 18 4.1. Grothendieck Ring is not a Domain 18 4.2. Grothendieck Ring of motives 19 5. APPENDIX : Tools for Birational geometry 20 Blowing Up 21 Resolution of Singularities 23 6. Glossary 26 References 30 1. Introduction First appeared in a letter of Grothendieck in Serre-Grothendieck cor- respondence (letter of 16/8/64), the Grothendieck ring of varieties is an interesting object lying at the heart of the theory of motivic inte- gration. A class of variety in this ring contains a lot of geometric information about the variety. For example, the topological euler characteristic, Hodge polynomials, Stably-birational properties, number of points if the variety is dened on a nite eld etc. Besides, the question of equality of these classes has given some important new results in bira- tional geometry (for example, Batyrev-Kontsevich's theorem). The Grothendieck Ring K0(V ark) is the quotient of the free abelian group generated by isomorphism classes of k-varieties, by the relation [XnY ] = [X]−[Y ], where Y is a closed subscheme of X; the ber prod- 0 0 uct over k induces a ring structure dened by [X]·[X ] = [(X ×k X )red]. Many geometric objects verify this kind of relations. It gives many re- alization maps, called additive invariants, containing some geometric information about the varieties.