Néron Models of Picard Varieties
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Divisors and the Picard Group
18.725 Algebraic Geometry I Lecture 15 Lecture 15: Divisors and the Picard Group Suppose X is irreducible. The (Weil) divisor DivW (X) is defined as the formal Z combinations of subvarieties of codimension 1. On the other hand, the Cartier divisor group, DivC (X), consists of subvariety locally given by a nonzero rational function defined up to multiplication by a nonvanishing function. Definition 1. An element of DivC (X) is given by 1. a covering Ui; and 2. Rational functions fi on Ui, fi 6= 0, ∗ such that on Ui \ Uj, fj = 'ijfi, where 'ij 2 O (Ui \ Uj). ∗ ∗ ∗ Another way to express this is that DivC (X) = Γ(K =O ), where K is the sheaf of nonzero rational functions, where O∗ is the sheaf of regular functions. Remark 1. Cartier divisors and invertible sheaves are equivalent (categorically). Given D 2 DivC (X), then we get an invertible subsheaf in K, locally it's fiO, the O-submodule generated by fi by construction it is locally isomorphic to O. Conversely if L ⊆ K is locally isomorphic to O, A system of local generators defines the data as above. Note that the abelian group structure on Γ(K∗=O∗) corresponds to multiplying by the ideals. ∗ ∗ ∗ ∗ Proposition 1. Pic(X) = DivC (X)=Im(K ) = Γ(K =O )= im Γ(K ). Proof. We already have a function DivC (X) = IFI ! Pic (IFI: invertible frational ideals) given by (L ⊆ K) 7! L. This map is an homomorphism. It is also onto: choosing a trivialization LjU = OjU gives an isomorphism ∼ ∗ ∗ L ⊗O⊇L K = K.No w let's look at its kernel: it consits of sections of K =O coming from O ⊆ K, which is just the same as the set of nonzero rational functions, which is im Γ(K∗) = Γ(K∗)=Γ(O∗). -
Rigid Analytic Curves and Their Jacobians
Rigid analytic curves and their Jacobians Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakult¨at f¨urMathematik und Wirtschaftswissenschaften der Universit¨atUlm vorgelegt von Sophie Schmieg aus Ebersberg Ulm 2013 Erstgutachter: Prof. Dr. Werner Lutkebohmert¨ Zweitgutachter: Prof. Dr. Stefan Wewers Amtierender Dekan: Prof. Dr. Dieter Rautenbach Tag der Promotion: 19. Juni 2013 Contents Glossary of Notations vii Introduction ix 1. The Jacobian of a curve in the complex case . ix 2. Mumford curves and general rigid analytic curves . ix 3. Outline of the chapters and the results of this work . x 4. Acknowledgements . xi 1. Some background on rigid geometry 1 1.1. Non-Archimedean analysis . 1 1.2. Affinoid varieties . 2 1.3. Admissible coverings and rigid analytic varieties . 3 1.4. The reduction of a rigid analytic variety . 4 1.5. Adic topology and complete rings . 5 1.6. Formal schemes . 9 1.7. Analytification of an algebraic variety . 11 1.8. Proper morphisms . 12 1.9. Etale´ morphisms . 13 1.10. Meromorphic functions . 14 1.11. Examples . 15 2. The structure of a formal analytic curve 17 2.1. Basic definitions . 17 2.2. The formal fiber of a point . 17 2.3. The formal fiber of regular points and double points . 22 2.4. The formal fiber of a general singular point . 23 2.5. Formal blow-ups . 27 2.6. The stable reduction theorem . 29 2.7. Examples . 31 3. Group objects and Jacobians 33 3.1. Some definitions from category theory . 33 3.2. Group objects . 35 3.3. Central extensions of group objects . -
Algebraic Cycles and Fibrations
Documenta Math. 1521 Algebraic Cycles and Fibrations Charles Vial Received: March 23, 2012 Revised: July 19, 2013 Communicated by Alexander Merkurjev Abstract. Let f : X → B be a projective surjective morphism between quasi-projective varieties. The goal of this paper is the study of the Chow groups of X in terms of the Chow groups of B and of the fibres of f. One of the applications concerns quadric bundles. When X and B are smooth projective and when f is a flat quadric fibration, we show that the Chow motive of X is “built” from the motives of varieties of dimension less than the dimension of B. 2010 Mathematics Subject Classification: 14C15, 14C25, 14C05, 14D99 Keywords and Phrases: Algebraic cycles, Chow groups, quadric bun- dles, motives, Chow–K¨unneth decomposition. Contents 1. Surjective morphisms and motives 1526 2. OntheChowgroupsofthefibres 1530 3. A generalisation of the projective bundle formula 1532 4. On the motive of quadric bundles 1534 5. Onthemotiveofasmoothblow-up 1536 6. Chow groups of varieties fibred by varieties with small Chow groups 1540 7. Applications 1547 References 1551 Documenta Mathematica 18 (2013) 1521–1553 1522 Charles Vial For a scheme X over a field k, CHi(X) denotes the rational Chow group of i- dimensional cycles on X modulo rational equivalence. Throughout, f : X → B will be a projective surjective morphism defined over k from a quasi-projective variety X of dimension dX to an irreducible quasi-projective variety B of di- mension dB, with various extra assumptions which will be explicitly stated. Let h be the class of a hyperplane section in the Picard group of X. -
SMOOTH PROJECTIVE SURFACES 1. Surfaces Definition 1.1. Let K Be
SMOOTH PROJECTIVE SURFACES SCOTT NOLLET Abstract. This talk will discuss some basics about smooth projective surfaces, leading to an open question. 1. Surfaces Definition 1.1. Let k be an algebraically closed field. A surface is a two dimensional n nonsingular closed subvariety S ⊂ Pk . Example 1.2. A few examples. (a) The simplest example is S = P2. (b) The nonsingular quadric surfaces Q ⊂ P3 with equation xw − yz = 0 has Picard group Z ⊕ Z generated by two opposite rulings. (c) The zero set of a general homogeneous polynomial f 2 k[x; y; z; w] of degree d gives rise to a smooth surface of degree d in P3. (d) If C and D are any two nonsingular complete curves, then each is projective (see previous lecture's notes), and composing with the Segre embedding we obtain a closed embedding S = C × D,! Pn. 1.1. Intersection theory. Given a surface S, let DivS be the group of Weil divisors. There is a unique pairing DivS × DivS ! Z denoted (C; D) 7! C · D such that (a) The pairing is bilinear. (b) The pairing is symmetric. (c) If C; D ⊂ S are smooth curves meeting transversely, then C · D = #(C \ D). (d) If C1 ∼ C2 (they are linearly equivalent), then C1 · D = C2 · D. Remark 1.3. When k = C one can describe the intersection pairing with topology. The divisor C gives rise to the line bundle L = OS(C) which has a first Chern class 2 c1(L) 2 H (S; Z) and similarly D gives rise to a class D 2 H2(S; Z) by triangulating the 0 ∼ components of D as real surfaces. -
Abelian Varieties
Abelian Varieties J.S. Milne Version 2.0 March 16, 2008 These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} } v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph shows the Tasman Glacier, New Zealand. Copyright c 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Introduction 1 I Abelian Varieties: Geometry 7 1 Definitions; Basic Properties. 7 2 Abelian Varieties over the Complex Numbers. 10 3 Rational Maps Into Abelian Varieties . 15 4 Review of cohomology . 20 5 The Theorem of the Cube. 21 6 Abelian Varieties are Projective . 27 7 Isogenies . 32 8 The Dual Abelian Variety. 34 9 The Dual Exact Sequence. 41 10 Endomorphisms . 42 11 Polarizations and Invertible Sheaves . 53 12 The Etale Cohomology of an Abelian Variety . 54 13 Weil Pairings . 57 14 The Rosati Involution . 61 15 Geometric Finiteness Theorems . 63 16 Families of Abelian Varieties . -
Nef Cones of Some Quot Schemes on a Smooth Projective Curve
NEF CONES OF SOME QUOT SCHEMES ON A SMOOTH PROJECTIVE CURVE CHANDRANANDAN GANGOPADHYAY AND RONNIE SEBASTIAN Abstract. Let C be a smooth projective curve over C. Let n; d ≥ 1. Let Q be the Quot scheme parameterizing torsion quotients of the vector n bundle OC of degree d. In this article we study the nef cone of Q. We give a complete description of the nef cone in the case of elliptic curves. We compute it in the case when d = 2 and C very general, in terms of the nef cone of the second symmetric product of C. In the case when n ≥ d and C very general, we give upper and lower bounds for the Nef cone. In general, we give a necessary and sufficient criterion for a divisor on Q to be nef. 1. Introduction Throughout this article we assume that the base field to be C. Let X 1 be a smooth projective variety and let N (X) be the R-vector space of R- divisors modulo numerical equivalence. It is known that N 1(X) is a finite dimensional vector space. The closed cone Nef(X) ⊂ N 1(X) is the cone of all R-divisors whose intersection product with any curve in X is non-negative. It has been an interesting problem to compute Nef(X). For example, when X = P(E) where E is a semistable vector bundle over a smooth projective curve, Miyaoka computed the Nef(X) in [Miy87]. In [BP14], Nef(X) was computed in the case when X is the Grassmann bundle associated to a vector bundle E on a smooth projective curve C, in terms of the Harder Narasimhan filtration of E. -
Algebraic Cycles on Generic Abelian Varieties Compositio Mathematica, Tome 100, No 1 (1996), P
COMPOSITIO MATHEMATICA NAJMUDDIN FAKHRUDDIN Algebraic cycles on generic abelian varieties Compositio Mathematica, tome 100, no 1 (1996), p. 101-119 <http://www.numdam.org/item?id=CM_1996__100_1_101_0> © Foundation Compositio Mathematica, 1996, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 100: 101-119,1996. 101 © 1996 KluwerAcademic Publishers. Printed in the Netherlands. Algebraic cycles on generic Abelian varieties NAJMUDDIN FAKHRUDDIN Department of Mathematics, the University of Chicago, Chicago, Illinois, USA Received 9 September 1994; accepted in final form 2 May 1995 Abstract. We formulate a conjecture about the Chow groups of generic Abelian varieties and prove it in a few cases. 1. Introduction In this paper we study the rational Chow groups of generic abelian varieties. More precisely we try to answer the following question: For which integers d do there exist "interesting" cycles of codimension d on the generic abelian variety of dimension g? By "interesting" cycles we mean cycles which are not in the subring of the Chow ring generated by divisors or cycles which are homologically equivalent to zero but not algebraically equivalent to zero. As background we recall that G. Ceresa [5] has shown that for the generic abelian variety of dimension three there exist codimen- sion two cycles which are homologically equivalent to zero but not algebraically equivalent to zero. -
The Homotopy Limit Problem and the Cellular Picard Group of Hermitian K-Theory
THE HOMOTOPY LIMIT PROBLEM AND THE CELLULAR PICARD GROUP OF HERMITIAN K-THEORY DREW HEARD Abstract. We use descent theoretic methods to solve the homotopy limit problem for Hermit- ian K-theory over quasi-compact and quasi-separated base schemes. As another application of these descent theoretic methods, we compute the cellular Picard group of 2-complete Hermit- ian K-theory over Spec(C), showing that the only invertible cellular spectra are suspensions of the tensor unit. 1. Introduction 1 2. Background 3 3. Some descent theory6 4. The cellular Picard group of Hermitian K-theory 10 References 14 Contents 1. Introduction Let KGL denote the motivic spectrum representing algebraic K-theory, along with its C2- action, and KQ the motivic spectrum representing Hermitian K-theory (both always taken over a fixed based scheme S). There is a map f : KQ KGL which is the motivic analog of the map f 0 : KO KU from real K-theory to topological! K-theory in stable homotopy. The purpose of this paper! is to investigate the ways in which f behaves like f 0. For example, there is an equivalence KO KUhC2 , and the homotopy limit problem in motivic homotopy ' hC theory asks if there is an equivalence KQ KGL 2 . For a field F , let vcd (F ) denote the ' 2 mod-2 cohomological dimension of the absolute Galois group of F (p 1). R¨ondigs{Spitzweck{ − Østvær [RSØ18] have shown that if F is a field of char(F ) = 2, and vcd2(F ) < , then the homotopy limit problem holds after η-completion over S =6 Spec(F ), that is, there1 is an equivalence KQ^ KGLhC2 . -
Chow Groupsand Derived Categories of K3 Surfaces
Current Developments in Algebraic Geometry MSRI Publications Volume 59, 2011 Chow groups and derived categories of K3 surfaces DANIEL HUYBRECHTS The geometry of a K3 surface (over C or over QN ) is reflected by its Chow group and its bounded derived category of coherent sheaves in different ways. The Chow group can be infinite dimensional over C (Mumford) and is expected to inject into cohomology over QN (Bloch–Beilinson). The derived category is difficult to describe explicitly, but its group of autoequivalences can be studied by means of the natural representation on cohomology. Conjecturally (Bridge- land) the kernel of this representation is generated by squares of spherical twists. The action of these spherical twists on the Chow ring can be determined explicitly by relating it to the natural subring introduced by Beauville and Voisin. 1. Introduction In algebraic geometry a K3 surface is a smooth projective surface X over a fixed 2 1 field K with trivial canonical bundle !X ' X and H .X; ᏻX / D 0. For us the field K will be either a number field, the field of algebraic numbers QN or the complex number field C. Nonprojective K3 surfaces play a central role in the theory of K3 surfaces and for some of the results that will be discussed in this text in particular, but here we will not discuss those more analytical aspects. An explicit example of a K3 surface is provided by the Fermat quartic in P3 4 C···C 4 given as the zero set of the polynomial x0 x3 . Kummer surfaces, i.e., minimal resolutions of the quotient of abelian surfaces by the sign involution, and elliptic K3 surfaces form other important classes of examples. -
Algebraic Geometry
University of Cambridge Mathematics Tripos Part III Algebraic Geometry Michaelmas, 2019 Lectures by M. Gross Notes by Qiangru Kuang Contents Contents 0 Introduction 2 0.1 Variety vs Scheme .......................... 2 0.2 Categorical philosophy ........................ 2 1 Sheaves 4 1.1 Sheafification ............................. 7 2 Schemes 10 2.1 Projective schemes .......................... 16 2.2 Open and closed subschemes .................... 19 2.3 Fibre products ............................ 20 3 Sheaves of OX -modules 25 3.1 Morphisms into projective space .................. 27 3.2 Divisors and the Picard groups ................... 30 3.3 Cartier divisor ............................ 34 4 Cohomology of sheaves 40 4.1 Čech cohomology ........................... 41 Index 53 1 0 Introduction 0 Introduction 0.1 Variety vs Scheme In classical algebraic geometry, we study varieties which are points where poly- nomials vanish. Why do we need schemes? Why not varieties? 1. With varieties, we always work with algebraically closed fields. Otherwise, the ideals are not really classical geometric objects. For example, consider I = (x2 + y2 + 1) ⊆ R[x; y]. V (I) = ;, I(V (I)) = R[x; y]. 2. Suppose one want to work on number theory. One is usually interested in n Diophantine equations, for example if I ⊆ Z[x1; : : : ; xn] then V (I) ⊆ Z . 2 2 3. Consider X1 = V (x − y ) ⊆ A ;X2 = V (x). Then 2 X1 \ X2 = V (x; x − y ): Consider I = (x; x − y2) = (x; y2) ⊆ k[x; y]. V (I) contains exactly one point, namely the origin, but the ideal I is not radical, reflecting the fact that X2 is the tangent to X1. Might it be reasonable to consider 2 k[x; y]=(x; y ) as the coordinate ring of X1 \ X2 rather than k[x; y]=(x; y)? Note y 2 k[x; y]=(x; y2) is non-zero but y2 = 0. -
The Picard Scheme of an Abelian Variety
Chapter VI. The Picard scheme of an abelian variety. 1. Relative Picard functors. § To place the notion of a dual abelian variety in its context, we start with a short discussion of relative Picard functors. Our goal is to sketch some general facts, without much discussion of proofs. Given a scheme X we write Pic(X)=H1(X, O∗ )= isomorphism classes of line bundles on X , X { } which has a natural group structure. (If τ is either the Zariski, or the ´etale, or the fppf topology 1 on Sch/X then we can also write Pic(X)=Hτ (X, Gm), viewing the group scheme Gm = Gm,X as a τ-sheaf on Sch/X ;seeExercise??.) If C is a complete non-singular curve over an algebraically closed field k then its Jacobian Jac(C) is an abelian variety parametrizing the degree zero divisor classes on C or, what is the same, the degree zero line bundles on C. (We refer to Chapter 14 for further discussion of Jacobians.) Thus, for every k K the degree map gives a homomorphism Pic(C ) Z, and ⊂ K → we have an exact sequence 0 Jac(C)(K) Pic(C ) Z 0 . −→ −→ K −→ −→ In view of the importance of the Jacobian in the theory of curves one may ask if, more generally, the line bundles on a variety X are parametrized by a scheme which is an extension of a discrete part by a connected group variety. If we want to study this in the general setting of a scheme f: X S over some basis S,we → are led to consider the contravariant functor P :(Sch )0 Ab given by X/S /S → P : T Pic(X )=H1(X T,G ) . -
Picard Numbers of Abelian Varieties and Related Arithmetic
Picard numbers of abelian varieties and related arithmetic Von der Fakultät für Mathematik und Physik der Gottfried Wilhelm Leibniz Universität Hannover zur Erlangung des Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von Roberto Laface geboren am 11. Juni 1990 in Reggio Calabria, Italien — 2016 — Referent: Prof. Dr. Matthias Schütt (Leibniz Universität Hannover) Korreferenten: Prof. Dr. François Charles (Université Paris-Sud, Paris, Frankreich) Prof. Dr. Lambertus van Geemen (Universitá degli Studi di Milano, Italien) Tag der Promotion: 28. Oktober 2016 Zusammenfassung Die vorliegende Arbeit befasst sich mit Picard-Zahlen von abelschen Varietäten und einigen verwandten arithmetischen Phänomenen. Eine Motivation hierfür bildet die Tatsache, dass singuläre abelsche Flächen, das heißt abelsche Flächen mit maximaler Picard-Zahl, mit einer zusätzlichen arithmetischen Struktur ausgestattet sind. Diese Struktur ist in dem transzendentalen Gitter kodiert und überträgt sich auf singuläre K3 Flächen mittels der Shioda-Inose Strukturen. Zuerst konzentrieren wir uns darauf, alle möglichen Zerlegungen einer singulären abelschen Fläche in ein Produkt von zueinander isogenen elliptischen Kurven mit komplexer Multiplikation zu finden. Hierbei nutzen wir eine Verallgemeinerung der Dirichlet Komposition, welche die Komposition von quadratischen Formen ver- schiedener Diskriminante ermöglicht. Unser Resultat ist eine Verallgemeinerung einer vorangegangenen Arbeit von Ma, der eine Formel für die Anzahl solcher Zerlegun- gen gefunden hat. Gleichzeitig liefert unser Ansatz einen alternativen Beweis für die Formel von Ma. Darauf aufbauend untersuchen wir den (relativen und absoluten) Körper der Moduli von singulären K3 Flächen. Wir verwenden eine Idee von Šafareviˇc,um unser Problem auf das entsprechende Problem über singuläre abelsche Flächen zu reduzieren. Dies ermöglicht uns die CM-Theorie von elliptischen Kurven anzuwenden, denn nach der Arbeit von Shioda und Mitani ist jede singuläre abelsche Fläche das Produkt von zwei solchen Kurven.