# 1 Existence of the Néron model

Size: px
Start display at page:

Transcription

1 Néron models Setting: S a Dedekind domain, K its field of fractions, A/K an abelian variety. A model of A/S is a flat, separable S-scheme of finite type X with X K = A. The nicest possible model over S is an S-abelian scheme (proper, smooth). Not every abelian variety admits such a model: if K is a p-adic field and S = O K, then A admitsanabelianschememodelovers iffthegaloisactionofγ K onh 1 ét (A,Q l)isunramified. Definition 1 A Neron model X of A over S is a smooth model that satisfies the Néron Mapping Property : For every smooth S-scheme Z and every map u K : Z K A,!v : Z X with v K = u K. Definition 2 A weak Néron model is a smooth model that satisfies the mapping property for every étale scheme Z. Remark 3 If the model exists, it is unique up to isomorphism by the universal property. The group structure of X K can be lifted to X by the mapping property. Proposition 4 If X is an S-abelian scheme, then X is the Néron model of its generic fiber. Proof. Let Z be any smooth S-scheme and u K : Z K X K be any map. By definition u is a rational map u : Z > X. Let V Z be the domain of definition of u. Clearly V contains Z K. Claim. s S closed point, V s is dense in Z s. Let ζ be a generic point of Z s. Then O Z,s is a DVR. By the valuative criterion for properness, there is a map Spec(O Z,ζ ) X. There exists U, an open neighborhood of ζ, such that u is defined on U. This implies the claim. The claim implies that the rational map u is well defined on a subscheme of codimension at most 1 (in fact, if dimx K = n, then dimx = n + 1 and u is well defined on a dense subscheme of Z K ). By a theorem of Weil, u is a morphism: Theorem 5 (Weil, BLR 4.4) Let S be a (not necessarily reduced) noetherian base scheme, u Z G an S-rational map from a smooth S-scheme Z to a smooth separable S-group scheme G. If u is well defined on a subscheme of codimension at most 1, then u is a morphism. As for the uniqueness, one has Theorem 6 (EGA 1, ) If we have two morphisms f 1,f 2 of S-schemes Z X, Z reduced, X separated. If there exists an open and dense subscheme U of Z s.t. f 1 and f 2 agree on U, then f 1 f 2. 1

2 1 Existence of the Néron model Let S be a Dedekind ring, K the field of fractions, A/K an abelian variety. We want to show that A admits a Néron model. Now A is a closed subscheme of a certain P n K, which in turn (canonically) embeds as an open subset of P n S. Take X to be the schematic closure of the inclusion A P n S. Then X is a proper model of A over S. Moreover, as X K Spec(K) is smooth, there exists an open subscheme S of S such that X S Spec(S ) is smooth. Now if for every point s S\S one can construct a Néron model of A over the local ring O S,s then we can just glue them together to get the global Néron model. Theorem 7 (BLR 1.2.4) For any model X/S of A, the following two conditions are equivalent: X is a Néron model; s S, X S O S,s is a Néron model. We are thus reduced to the local case. From now on, S = Spec(R), where R is a DVR, K is the fraction field, and k is the residue field. Let k s be the separable closure of k. Let R sh be the strict henselianization of R, i.e. 1.1 Plan of proof We want to constuct a sequence of models R sh = lim R R /R étale k /k k s /k X 0 X 1 X 2 X 3 X 4 where X 0 is a proper model of A, X 1 is obtained from X 0 by blowing up some components of the special fiber, X 2 is the smooth locus of X 1 (and is a weak Néron model), X 3 is obtained from X 2 by removing some components (and has a birational group structure), and X 4 is obtained from X 3 by Weil s theorem that extends the birational group structure to a group structure. Now X 0 is proper, and X 0,k is proper and geometrically connected (EGA IV ). However, X 0,k is in general neither smooth nor reduced. Aim: recover smoothness dropping properness Definition 8 Let X be a flat R-scheme of finite type with smooth generic fiber. A smoothening of X is a proper map X X such that: (1) X K = X K ; (2) for every étale extension R /R, X smooth (R ) X(R ) is surjective; or equivalently, (2 ) X smooth (Rsh ) X(R sh ). Theorem 9 X as above. A smoothening always exists. 2

3 Let X 0 be a proper model of A, X 1 X 0 a smoothening of X 0. Then we can take X 2 to be the smooth locus of X 1. Proposition 10 X 2 is a weak Néron model of A. Proof. By definition of a smoothening (specifically, property 2), every étale map lifts to X 2. Remark 11 A weak Néron model is not unique. 1.2 Existence of smoothenings The idea is to blow up some components of X K. Definition 12 (measure for defect of smoothness) X as above. For any a : Spec(R sh ) X we set δ(a) to be the length of the torsion submodule of a Ω 1 X/R. For any subset E of X(R sh ) we denote δ(x,e) = max{δ(a),a E} Theorem 13 (BLR 3.3 prop 3) δ(a) is bounded as a varies in X(R sh ) Lemma 14 δ(a) = 0 iff a factors through the smooth locus of X. Proof. : if a is smooth, then locally around a the sheaf Ω 1 X/R is free of finite type, hence a Ω 1 X/R is free : if a Ω 1 X/R = 0, d k denotes the dimension of X k, and d K denotes the dimension of X K, then by semi-continuity d k d K. As R sh is a DVR, the pullback a Ω 1 X/R is free (since it is torsion-free of finite type), so it is generated by d K elements. But then a k Ω1 X k /k is generated by d K elements, and d K d k, which implies that - locally around a k - X k /k is smooth of dimension d K. Since X/R is flat and locally X K /K and X k /k are smooth, X/R is smooth Canonical partition Given X as above and E X(R sh ), we will define a collection of geometrically reduced k-subschemes Y 1,U 1,Y 2,U 2,...,Y t,u t of X k and a partition E = E 1 E 2... E t. Define π to be the specialization map on R sh, i.e. π : X(R sh ) X k (k s ). 1. Let Y 1 be the Zariski closure of π(e) in X k 2. Let U 1 be the largest subscheme of Y such that U 1 is smooth; Ω 1 X/K Y 1 is locally free 3. E 1 := π 1 (U 1 (k s )) E. Note that Y 1 is geometrically reduced and that U 1 is dense in Y 1. Now suppose we have defined E 1,...,E i for i 1. Apply this construction to E \ (E 1... E i ) to get a closed subscheme Y i+1, an open dense U i+1, and a subset E i+1. This process will terminate in finitely many steps as X k is Noetherian. We denote t = t(x,e) the length of this partition ( length of partition of E in X ). 3

4 1.2.2 End of proof Lemma 15 (BLR 3.3.5) X,E as above. Let a E be such that a k = π(a) is a singular point of X k. Assume that a E i (i.e. a U i (k s )). Let X X be the blow-up along Y i and a be the unique lift of a to X (existence and uniqueness of the lift follow from the valuation criterion for properness and the fact that X X is proper). Then δ(a ) {0,δ(a) 1}. Let X,E as above. Denote E k = π(e). Definition 16 Let Y be a closed subscheme of X k. We say that it is E-permissible if Y is geometrically reduced and F = Y(k s ) E k satisfies: 1. F lies in Y smooth 2. F lies in the largest subscheme of Y over which Ω 1 X/R is locally free 3. F is dense in Y In case Y is permissible, we say that a blow-up X X along Y is E-permissible. Theorem 17 X,E as above. There is a X X that is a finite sequence of E-permissible blow-ups such that for any a E, there exists a unique lift of a to a point a in X sm(r sh ). Remark 18 The smoothening theorem is the case E = R sh Proof. By induction on δ(x,e)+t(x,e) 1. If δ(x,e) = 0, then all the points of X are smooth, and there is nothing to do. So we can assume δ(x,e) 1. Consider the canonical partition E = E 1... E t and let X X be the E-permissible blowup along Y t. Recall that by construction we have π(e t ) U t (k s ) Y t (k s ), and that π(e t ) are dense in Y t. By the lemma, for every a in E t the unique lift a of a satisfies δ(a ) < δ(a), and therefore δ(x,e t ) < δ(x,e t ). By induction, there is a finite chain of blow-ups such that E t lifts uniquely to X smooth. If t = 1 we are done. If t > 1, consider E = E \E t on X ; E has a canonical partition which is again given by E 1,...,E t 1, and t(x,e ) < t(x,e). Then induct. 4

5 2 Part 2 - from weak models to strong ones Recall that we are working with S = Spec(R), R a DVR, and K the field of fractions of R. Let X be a weak Néron model of A. 2.1 Invariant differential forms and minimal components Let S be a scheme, G/S a smooth S-group scheme of relative dimension d. The sheaf Ω d G/S := Λd Ω 1 G/S is invertible, generated by a (left)-invariant differential form. Choose such a form ω for A/K. As A/K is proper, hence commutative, ω is also rightinvariant. Let X/R be a smooth model of A, of relative dimension d. The sheaf Ω d X/R is again invertible, and satisfies Ω d X/R R K = Ω d A/K. Now for every irreducible component W of X k let η be the generic point and note that O X,η is a DVR, with the same uniformizer π as R. Hence there exists an r Z such that Ω d X/R is generated by π r ω on O X,η. We set ord W (ω) = r and ρ := min W ord W(ω). Replacing ω by π ρ ω we can suppose ρ = 0. We say that a component of X is minimal if ord W (ω) = 0. We take X 3 (our next step in constructing a Néron model) to be X minus the non-minimal components. Lemma 19 In this case (i.e. ρ = 0), ω extends to a section of Ω d X/R, and it is a generator of Ω d X 3 /R. Proof. 1. As X is normal, if ω is well-defined in codimension at most 1 it extends to a global section. 2. The zero locus of a section of a line bundle on an integral scheme is of pure codimension one. 3. ω X3 does not vanish on codimension 1, so ω X3 is everywhere nonvanishing (hence a generator). 2.2 Properties of the weak Néron model Definition 20 Let S be a scheme and X,Y be flat schemes of finite type over S. 1. We say that an open set U X is S-dense if s S the fiber U s is schematically dense in X s. 5

6 2. An S-rational map f : X > Y is an equivalence class of maps U Y where U is an S-dense subscheme of X. The equivalence relation is given by (f : U Y) (g : V Y) if and only if W dense in U V such that f W = g W. Remark 21 If Y is separated and X is irreducible there exists a maximal S-dense open subscheme of X such that f is well-defined on U. This U is called the domain of definition of f. Theorem 22 Let X be a weak Néron model of A. Then X satisfies the R-rational Néron mapping property, namely smooth R-scheme Z, u K : Z K > A,!R rational map f : Z > X such that f K = u K. Proof. Preliminary reductions: 1. We can suppose that Z is of finite type over R (at any rate, the property is local). 2. We can reduce to the case when Z has only one irreducible component in the special fiber: as Z k is smooth over k, Z k = irreducible components. 3. Let W Z be the domain of definition of u K. Take Y the schematic closure of Z k \W in Z. Then we only have to work over Z\Y, i.e. we can suppose that u K is a morphism. Note that Z \Y is R-dense in Z. So (without loss of generality) we have a morphism u K : Z K A. Let Γ K be the closed subscheme of Z K A given by the graph of u K, and take Γ to be the schematic closure of Γ K in Z R X. Let p : Γ Z be the projection. It is enough to prove that p admits a section on an R-dense, open subscheme of Z. Claim: p k : Γ k (k s ) Z k (k s ) is surjective. Proof. Any point z k Z k (k s ) lifts to a z Z(R sh ) since Z/R is smooth and R sh is Henselian. Let z K be the generic point of Z. Consider the map u K : z K Z K A. By the weak Néron property, u K extends to a map u : Spec(R sh ) X. Now u K (Z K ) = X K is a generic point of u(z) = X. We deduce from this a point (Z,X) of Γ, such that (Z k,x k ) projects to z k via p k. Now p k (Γ k ) is a constructible subset of X k. The projection pi k (Γ k ) contains all the k s - points, so it contains a dense open subscheme of Z K. On the other hand, pi k is dominated: for every η Z k, ζ Γ such that π(ζ) = η. Consider the map of DVRs ϕ : O Z,η O Γ,ζ. The map ϕ K is an isomorphism, since p K : Γ K Z K has a section. Hence ϕ is an isomorphism, and by descent theory ϕ induces an isomorphism U V, where U and V are open neighborhoods of η,ζ respectively. This implies that p : Γ Z has a section on U, and as U contains the generic point of Z it is also R-dense in it. 6

7 2.3 Properties of X 3 Theorem 23 m K : A A A, the group law on A, can be extended to an R-birational group law m : X 3 X 3 > X 3 Remark 24 An R-birational group law is a map that satisfies is R-birational; is R-birational. Φ : X 3 X 3 > X 3 X 3 (x, y) (x, m(x, y)) Ψ : X 3 X 3 > X 3 X 3 (x, y) (m(x, y), y) Proof. The properties of the weak Néron model (previous section) imply the existence of a R-rational lift m : X 3 X 3 > X of m K. Let D X 3 R X 3 be the domain of definition of m, and let ϕ : D X 3 R X (x,y) (x,m(x,y)). Let ω be an invariant differential form on Ω d X/R and ω = pr2 ω Ωd X 3 X/R. We have ϕ ω = ω since this is true on the generic fiber. Let ξ = (α,γ) be a generic point of X 3 X 3. We want to show that η = ϕ(ξ) is in X 3 X 3, so that we can consider ϕ as a birational map with image in X 3 X 3. Let r = ord γ (ω) = ord η (ω) 0. The form π r ω is a generator of Ω d D/X 3 in O D,η. On the other hand, ϕ (π r ω ) = bω is also a generator (b O D,ξ ). By comparing orders, r = 0 and b invertible, so (by definition of a minimal component) γ X 3,k and likewise η X 3,k X 3,k since ord η (ω ) = 0. Therefore ϕ maps (X 3 X 3 ) k to (X 3 X 3 ) k. Let U = D (X 3 X 3 ). Then ϕ induces a morphism U X 3 X 3, and we can define m = pr 2 ϕ : U X 3 X 3 X 3. We can also define Ψ in a similar way. To show that Φ is birational, one needs to show that ϕ induces an isomorphism of U onto an open subscheme of X 3 X 3. To do so, simply notice that ϕ is an isomorphism between Ω d X 3 X 3 /X 3 and Ω d U/X 3, hence it is also an isomorphism between the Ω 1 s. It follows from the diagram U ϕ X 3 X 3 smooth X 3 smooth that ϕ is étale (since the schemes are smooth over the base and ϕ is an isomorphism on Ω 1 ). By Zariski s main theorem, α factors through an open immersion U Y and a finite map Y X 3 X 3. But since the finite map is an isomorphism on the generic fiber, it is an isomorphism of Y onto an open subscheme of X 3 X 3, as desired. 7

8 Theorem 25 (BLR 5, 6) Given X 3,A there exists X 4 (smooth separable R-group scheme of finite type) containing X 3 as an R-dense open subscheme with the following properties: (X 4 ) K = A; the group law of X 4 extends the R-birational group law on X 3. X 4 is unique, and is a Néron model. 2.4 Néron models of elliptic curves K a local field, R its ring of integers, k the residue field. Definition 26 A K-curve is a proper, geometrically connected scheme over K of dimension 1 An R-curve is a proper flat R-scheme whose geometric fibers are connected Theorem 27 (Resolution of singularities) If C is a smooth K-curve there exists a regular R-model of C. Proof. By blowing up. A proof can be found in [?]. Definition 28 Given a smooth K-curve C of genus at least 1, a minimal regular R-model of C is a regular R-model X of C such that for any other regular R-model Y of C, there exists a morphism Y X which induces the identity on the generic fibers. Theorem 29 [?, XXX] C as above. There exists a (unique) minimal regular R-model of C. Let now E/K be an elliptic curve and E be its minimal regular model. Consider the smooth locus of E. Since (E sm ) K = E admits a map (the identity) towards E, by the Néron mapping property there exists a (canonical) map E sm Néron(E). Theorem 30 (BLR 1.5 proposition 1) This canonical map is an isomorphism. 2.5 Néron model of a Jacobian Let S be any scheme and X/S be an S-scheme. We can define the relative Picard functor by P X/S : S Sch Set T Pic(X S T). The functor Pic X/S is the sheafification of P X/S for the fppf topology on S Sch. We have Pic X/S (T) = H 0 (T,R 1 f T, G m ), where f is the structure map X T T. If C is a smooth curve over K, then Pic C/K is a smooth commutative group scheme over K. Its connected component is an abelian variety over K (the Jacobian of C). if X is an R-model of C that is also semi-stable (i.e. X k is reduced with at most ordinary double points as singularities), then Pic X/S is representable by an R-group scheme, whose connected component Pic 0 X/S is a semistable model of Jac(C). By the Néron mapping property, there is a map Pic 0 X/S Néron(Jac(C)), and we have 8

9 Theorem 31 (BLR 9.7) This induces a canonical isomorphism Pic 0 X/S Néron(Jac(C))0. Remark 32 If k is perfect, then C has potentially semistable reduction. if C has semistable reduction, then the minimal regular model of C is semistable 9

### Néron models of abelian varieties

Néron models of abelian varieties Matthieu Romagny Summer School on SGA3, September 3, 2011 Abstract : We present a survey of the construction of Néron models of abelian varieties, as an application of

### NÉRON MODELS. Contents

NÉRON MODELS SAM LICHTENSTEIN Contents 1. Introduction: Definition and First Properties 1 2. First example: abelian schemes 6 3. Sketch of proof of Theorem 1.3.10 11 4. Elliptic curves 16 5. Some more

### Elliptic curves, Néron models, and duality

Elliptic curves, Néron models, and duality Jean Gillibert Durham, Pure Maths Colloquium 26th February 2007 1 Elliptic curves and Weierstrass equations Let K be a field Definition: An elliptic curve over

### where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset

Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring

### APPENDIX 3: AN OVERVIEW OF CHOW GROUPS

APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss

### LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for non-stable-rationality based on the decomposition of the diagonal in the Chow group. This criterion

### CANONICAL EXTENSIONS OF NÉRON MODELS OF JACOBIANS

CANONICAL EXTENSIONS OF NÉRON MODELS OF JACOBIANS BRYDEN CAIS Abstract. Let A be the Néron model of an abelian variety A K over the fraction field K of a discrete valuation ring R. Due to work of Mazur-Messing,

### OFER GABBER, QING LIU, AND DINO LORENZINI

PERIOD, INDEX, AND AN INVARIANT OF GROTHENDIECK FOR RELATIVE CURVES OFER GABBER, QING LIU, AND DINO LORENZINI 1. An invariant of Grothendieck for relative curves Let S be a noetherian regular connected

### HYPERSURFACES IN PROJECTIVE SCHEMES AND A MOVING LEMMA

HYPERSURFACES IN PROJECTIVE SCHEMES AND A MOVING LEMMA OFER GABBER, QING LIU, AND DINO LORENZINI Abstract. Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique

### 1.6.1 What are Néron Models?

18 1. Abelian Varieties: 10/20/03 notes by W. Stein 1.6.1 What are Néron Models? Suppose E is an elliptic curve over Q. If is the minimal discriminant of E, then E has good reduction at p for all p, in

### MINIMAL MODELS FOR ELLIPTIC CURVES

MINIMAL MODELS FOR ELLIPTIC CURVES BRIAN CONRAD 1. Introduction In the 1960 s, the efforts of many mathematicians (Kodaira, Néron, Raynaud, Tate, Lichtenbaum, Shafarevich, Lipman, and Deligne-Mumford)

### Artin Approximation and Proper Base Change

Artin Approximation and Proper Base Change Akshay Venkatesh November 9, 2016 1 Proper base change theorem We re going to talk through the proof of the Proper Base Change Theorem: Theorem 1.1. Let f : X

### MATH 233B, FLATNESS AND SMOOTHNESS.

MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)

### Smooth morphisms. Peter Bruin 21 February 2007

Smooth morphisms Peter Bruin 21 February 2007 Introduction The goal of this talk is to define smooth morphisms of schemes, which are one of the main ingredients in Néron s fundamental theorem [BLR, 1.3,

### 1 Notations and Statement of the Main Results

An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main

### Notes on p-divisible Groups

Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete

### Proof of the Shafarevich conjecture

Proof of the Shafarevich conjecture Rebecca Bellovin We have an isogeny of degree l h φ : B 1 B 2 of abelian varieties over K isogenous to A. We wish to show that h(b 1 ) = h(b 2 ). By filtering the kernel

### Topics in Algebraic Geometry

Topics in Algebraic Geometry Nikitas Nikandros, 3928675, Utrecht University n.nikandros@students.uu.nl March 2, 2016 1 Introduction and motivation In this talk i will give an incomplete and at sometimes

### Tunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society

unisian Journal of Mathematics an international publication organized by the unisian Mathematical Society Ramification groups of coverings and valuations akeshi Saito 2019 vol. 1 no. 3 msp msp UNISIAN

### NOTES ON ABELIAN VARIETIES

NOTES ON ABELIAN VARIETIES YICHAO TIAN AND WEIZHE ZHENG We fix a field k and an algebraic closure k of k. A variety over k is a geometrically integral and separated scheme of finite type over k. If X and

### ÉTALE π 1 OF A SMOOTH CURVE

ÉTALE π 1 OF A SMOOTH CURVE AKHIL MATHEW 1. Introduction One of the early achievements of Grothendieck s theory of schemes was the (partial) computation of the étale fundamental group of a smooth projective

### Néron Models of Elliptic Curves.

Néron Models of Elliptic Curves. Marco Streng 5th April 2007 These notes are meant as an introduction and a collection of references to Néron models of elliptic curves. We use Liu [Liu02] and Silverman

### Semi-factorial nodal curves and Néron models of jacobians

Semi-factorial nodal curves and Néron models of jacobians arxiv:602.03700v2 [math.ag] 24 Oct 206 Giulio Orecchia email: g.orecchia@math.leidenuniv.nl Mathematisch Instituut Leiden, Niels Bohrweg, 2333CA

### 1.5.4 Every abelian variety is a quotient of a Jacobian

16 1. Abelian Varieties: 10/10/03 notes by W. Stein 1.5.4 Every abelian variety is a quotient of a Jacobian Over an infinite field, every abelin variety can be obtained as a quotient of a Jacobian variety.

### Chern classes à la Grothendieck

Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces

### THE SMOOTH BASE CHANGE THEOREM

THE SMOOTH BASE CHANGE THEOREM AARON LANDESMAN CONTENTS 1. Introduction 2 1.1. Statement of the smooth base change theorem 2 1.2. Topological smooth base change 4 1.3. A useful case of smooth base change

### 1. Algebraic vector bundles. Affine Varieties

0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

### HARTSHORNE EXERCISES

HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing

### SEMISTABLE REDUCTION FOR ABELIAN VARIETIES

SEMISTABLE REDUCTION FOR ABELIAN VARIETIES BRIAN CONRAD 1. Introduction The semistable reduction theorem for curves was discussed in Christian s notes. In these notes, we will use that result to prove

### mult V f, where the sum ranges over prime divisor V X. We say that two divisors D 1 and D 2 are linearly equivalent, denoted by sending

2. The canonical divisor In this section we will introduce one of the most important invariants in the birational classification of varieties. Definition 2.1. Let X be a normal quasi-projective variety

### Math 248B. Applications of base change for coherent cohomology

Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

### A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1

A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Proper morphisms 1 2. Scheme-theoretic closure, and scheme-theoretic image 2 3. Rational maps 3 4. Examples of rational maps 5 Last day:

### INERTIA GROUPS AND FIBERS

INERTIA GROUPS AND FIBERS BRIAN CONRAD Let K be a global field and X, Y two proper, connected K-schemes, with X normal and Y regular. Let f : X Y be a finite, flat, generically Galois K-morphism which

### On Rational Points of Varieties over Complete Local Fields with Algebraically Closed Residue Field

On Rational Points of Varieties over Complete Local Fields with Algebraically Closed Residue Field Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.)

### Preliminary Exam Topics Sarah Mayes

Preliminary Exam Topics Sarah Mayes 1. Sheaves Definition of a sheaf Definition of stalks of a sheaf Definition and universal property of sheaf associated to a presheaf [Hartshorne, II.1.2] Definition

### 3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).

3. Lecture 3 3.1. Freely generate qfh-sheaves. We recall that if F is a homotopy invariant presheaf with transfers in the sense of the last lecture, then we have a well defined pairing F(X) H 0 (X/S) F(S)

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48 RAVI VAKIL CONTENTS 1. A little more about cubic plane curves 1 2. Line bundles of degree 4, and Poncelet s Porism 1 3. Fun counterexamples using elliptic curves

### MOISHEZON SPACES IN RIGID GEOMETRY

MOISHEZON SPACES IN RIGID GEOMETRY BRIAN CONRAD Abstract. We prove that all proper rigid-analytic spaces with enough algebraically independent meromorphic functions are algebraic (in the sense of proper

### 6. Lecture cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in Suslin-Voevodsky s approach to not

6. Lecture 6 6.1. cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in Suslin-Voevodsky s approach to not only motivic cohomology, but also to Morel-Voevodsky

### ORDINARY VARIETIES AND THE COMPARISON BETWEEN MULTIPLIER IDEALS AND TEST IDEALS

ORDINARY VARIETIES AND THE COMPARISON BETWEEN MULTIPLIER IDEALS AND TEST IDEALS MIRCEA MUSTAŢĂ AND VASUDEVAN SRINIVAS Abstract. We consider the following conjecture: if X is a smooth n-dimensional projective

### MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

### The Picard Scheme and the Dual Abelian Variety

The Picard Scheme and the Dual Abelian Variety Gabriel Dorfsman-Hopkins May 3, 2015 Contents 1 Introduction 2 1.1 Representable Functors and their Applications to Moduli Problems............... 2 1.2 Conditions

### Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

### THE KEEL MORI THEOREM VIA STACKS

THE KEEL MORI THEOREM VIA STACKS BRIAN CONRAD 1. Introduction Let X be an Artin stack (always assumed to have quasi-compact and separated diagonal over Spec Z; cf. [2, 1.3]). A coarse moduli space for

### Section Blowing Up

Section 2.7.1 - Blowing Up Daniel Murfet October 5, 2006 Now we come to the generalised notion of blowing up. In (I, 4) we defined the blowing up of a variety with respect to a point. Now we will define

### VERY STABLE BUNDLES AND PROPERNESS OF THE HITCHIN MAP

VERY STABLE BUNDLES AND PROPERNESS OF THE HITCHIN MAP CHRISTIAN PAULY AND ANA PEÓN-NIETO Abstract. Let X be a smooth complex projective curve of genus g 2 and let K be its canonical bundle. In this note

### CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES

CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES BRIAN OSSERMAN The purpose of this cheat sheet is to provide an easy reference for definitions of various properties of morphisms of schemes, and basic results

### MATH 8254 ALGEBRAIC GEOMETRY HOMEWORK 1

MATH 8254 ALGEBRAIC GEOMETRY HOMEWORK 1 CİHAN BAHRAN I discussed several of the problems here with Cheuk Yu Mak and Chen Wan. 4.1.12. Let X be a normal and proper algebraic variety over a field k. Show

### RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES RIMS A Note on an Anabelian Open Basis for a Smooth Variety. Yuichiro HOSHI.

RIMS-1898 A Note on an Anabelian Open Basis for a Smooth Variety By Yuichiro HOSHI January 2019 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan A Note on an Anabelian Open Basis

### DEFORMATIONS VIA DIMENSION THEORY

DEFORMATIONS VIA DIMENSION THEORY BRIAN OSSERMAN Abstract. We show that standard arguments for deformations based on dimension counts can also be applied over a (not necessarily Noetherian) valuation ring

### Abelian Varieties and the Fourier Mukai transformations (Foschungsseminar 2005)

Abelian Varieties and the Fourier Mukai transformations (Foschungsseminar 2005) U. Bunke April 27, 2005 Contents 1 Abelian varieties 2 1.1 Basic definitions................................. 2 1.2 Examples

### ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open

### NOTES ON THE CONSTRUCTION OF THE MODULI SPACE OF CURVES

NOTES ON THE CONSTRUCTION OF THE MODULI SPACE OF CURVES DAN EDIDIN The purpose of these notes is to discuss the problem of moduli for curves of genus g 3 1 and outline the construction of the (coarse)

### Wild ramification and the characteristic cycle of an l-adic sheaf

Wild ramification and the characteristic cycle of an l-adic sheaf Takeshi Saito March 14 (Chicago), 23 (Toronto), 2007 Abstract The graded quotients of the logarithmic higher ramification groups of a local

### Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

### Synopsis of material from EGA Chapter II, 5

Synopsis of material from EGA Chapter II, 5 5. Quasi-affine, quasi-projective, proper and projective morphisms 5.1. Quasi-affine morphisms. Definition (5.1.1). A scheme is quasi-affine if it is isomorphic

### AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES

AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES YEHAO ZHOU Conventions In this lecture note, a variety means a separated algebraic variety over complex numbers, and sheaves are C-linear. 1.

### Motivic integration on Artin n-stacks

Motivic integration on Artin n-stacks Chetan Balwe Nov 13,2009 1 / 48 Prestacks (This treatment of stacks is due to B. Toën and G. Vezzosi.) Let S be a fixed base scheme. Let (Aff /S) be the category of

### where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism

8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the

### Appendix by Brian Conrad: The Shimura construction in weight 2

CHAPTER 5 Appendix by Brian Conrad: The Shimura construction in weight 2 The purpose of this appendix is to explain the ideas of Eichler-Shimura for constructing the two-dimensional -adic representations

### UNIVERSAL LOG STRUCTURES ON SEMI-STABLE VARIETIES

UNIVERSAL LOG STRUCTURES ON SEMI-STABLE VARIETIES MARTIN C. OLSSON Abstract. Fix a morphism of schemes f : X S which is flat, proper, and fiber-by-fiber semi-stable. Let IV LS be the functor on the category

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### CHEVALLEY S THEOREM AND COMPLETE VARIETIES

CHEVALLEY S THEOREM AND COMPLETE VARIETIES BRIAN OSSERMAN In this note, we introduce the concept which plays the role of compactness for varieties completeness. We prove that completeness can be characterized

### Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].

Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.

### 14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski

14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski topology are very large, it is natural to view this as

### Zero-cycles on surfaces

Erasmus Mundus ALGANT Master thesis Zero-cycles on surfaces Author: Maxim Mornev Advisor: Dr. François Charles Orsay, 2013 Contents 1 Notation and conventions 3 2 The conjecture of Bloch 4 3 Algebraic

### DIVISOR CLASSES AND THE VIRTUAL CANONICAL BUNDLE FOR GENUS 0 MAPS

DIVISOR CLASSES AND THE VIRTUAL CANONICAL BUNDLE FOR GENUS 0 MAPS A. J. DE JONG AND JASON STARR Abstract. We prove divisor class relations for families of genus 0 curves and used them to compute the divisor

### THE SHIMURA-TANIYAMA FORMULA AND p-divisible GROUPS

THE SHIMURA-TANIYAMA FORMULA AND p-divisible GROUPS DANIEL LITT Let us fix the following notation: 1. Notation and Introduction K is a number field; L is a CM field with totally real subfield L + ; (A,

### arxiv:alg-geom/ v1 21 Mar 1996

AN INTERSECTION NUMBER FOR THE PUNCTUAL HILBERT SCHEME OF A SURFACE arxiv:alg-geom/960305v 2 Mar 996 GEIR ELLINGSRUD AND STEIN ARILD STRØMME. Introduction Let S be a smooth projective surface over an algebraically

### 1 Flat, Smooth, Unramified, and Étale Morphisms

1 Flat, Smooth, Unramified, and Étale Morphisms 1.1 Flat morphisms Definition 1.1. An A-module M is flat if the (right-exact) functor A M is exact. It is faithfully flat if a complex of A-modules P N Q

### ON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE

ON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE DILETTA MARTINELLI, STEFAN SCHREIEDER, AND LUCA TASIN Abstract. We show that the number of marked minimal models of an n-dimensional

### Artin-Tate Conjecture, fibered surfaces, and minimal regular proper model

Artin-Tate Conjecture, fibered surfaces, and minimal regular proper model Brian Conrad November 22, 2015 1 Minimal Models of Surfaces 1.1 Notation and setup Let K be a global function field of characteristic

### 10. Smooth Varieties. 82 Andreas Gathmann

82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It

### REFINED ALTERATIONS BHARGAV BHATT AND ANDREW SNOWDEN

REFINED ALTERATIONS BHARGAV BHATT AND ANDREW SNOWDEN 1. INTRODUCTION Fix an algebraic variety X over a field. Hironaa showed [Hir64] that if has characteristic 0, then one can find a proper birational

### CHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998

CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic

### Néron models of jacobians over base schemes of dimension greater than 1

Néron models of jacobians over base schemes of dimension greater than 1 David Holmes arxiv:1402.0647v4 [math.ag] 26 Feb 2016 February 29, 2016 Abstract We investigate to what extent the theory of Néron

### Rigid Geometry and Applications II. Kazuhiro Fujiwara & Fumiharu Kato

Rigid Geometry and Applications II Kazuhiro Fujiwara & Fumiharu Kato Birational Geometry from Zariski s viewpoint S U D S : coherent (= quasi-compact and quasi-separated) (analog. compact Hausdorff) U

### NONSINGULAR CURVES BRIAN OSSERMAN

NONSINGULAR CURVES BRIAN OSSERMAN The primary goal of this note is to prove that every abstract nonsingular curve can be realized as an open subset of a (unique) nonsingular projective curve. Note that

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41 RAVI VAKIL CONTENTS 1. Normalization 1 2. Extending maps to projective schemes over smooth codimension one points: the clear denominators theorem 5 Welcome back!

### Lecture 3: Flat Morphisms

Lecture 3: Flat Morphisms September 29, 2014 1 A crash course on Properties of Schemes For more details on these properties, see [Hartshorne, II, 1-5]. 1.1 Open and Closed Subschemes If (X, O X ) is a

### Weight functions on non-archimedean analytic spaces and the Kontsevich Soibelman skeleton

Algebraic Geometry 2 (3) (2015) 365 404 doi:10.14231/ag-2015-016 Weight functions on non-archimedean analytic spaces and the Kontsevich Soibelman skeleton Mircea Mustaţă and Johannes Nicaise Abstract We

### ON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE

ON THE NUMBER AND BOUNDEDNESS OF LOG MINIMAL MODELS OF GENERAL TYPE DILETTA MARTINELLI, STEFAN SCHREIEDER, AND LUCA TASIN Abstract. We show that the number of marked minimal models of an n-dimensional

### Quasi-compactness of Néron models, and an application to torsion points

Quasi-compactness of Néron models, and an application to torsion points David Holmes arxiv:1604.01155v2 [math.ag] 7 Apr 2016 April 8, 2016 Abstract We prove that Néron models of jacobians of generically-smooth

### SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY

SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY MARC LEVINE Contents 0. Introduction 1 1. The category of schemes 2 1.1. The spectrum of a commutative ring 2 1.2. Ringed spaces 5 1.3. Schemes 10 1.4. Schemes

### An Atlas For Bun r (X)

An Atlas For Bun r (X) As told by Dennis Gaitsgory to Nir Avni October 28, 2009 1 Bun r (X) Is Not Of Finite Type The goal of this lecture is to find a smooth atlas locally of finite type for the stack

### Remarks on the existence of Cartier divisors

arxiv:math/0001104v1 [math.ag] 19 Jan 2000 Remarks on the existence of Cartier divisors Stefan Schröer October 22, 2018 Abstract We characterize those invertible sheaves on a noetherian scheme which are

### FALTINGS SEMINAR TALK. (rf)(a) = rf(a) = f(ra)

FALTINGS SEMINAR TALK SIMON RUBINSTEIN-SALZEDO 1. Cartier duality Let R be a commutative ring, and let G = Spec A be a commutative group scheme, where A is a finite flat algebra over R. Let A = Hom R mod

### Raynaud on F -vector schemes and prolongation

Raynaud on F -vector schemes and prolongation Melanie Matchett Wood November 7, 2010 1 Introduction and Motivation Given a finite, flat commutative group scheme G killed by p over R of mixed characteristic

### 12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n

12. Linear systems Theorem 12.1. Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s

### (1) is an invertible sheaf on X, which is generated by the global sections

7. Linear systems First a word about the base scheme. We would lie to wor in enough generality to cover the general case. On the other hand, it taes some wor to state properly the general results if one

### COMPLEX ALGEBRAIC SURFACES CLASS 9

COMPLEX ALGEBRAIC SURFACES CLASS 9 RAVI VAKIL CONTENTS 1. Construction of Castelnuovo s contraction map 1 2. Ruled surfaces 3 (At the end of last lecture I discussed the Weak Factorization Theorem, Resolution

### Zariski s Main Theorem and some applications

Zariski s Main Theorem and some applications Akhil Mathew January 18, 2011 Abstract We give an exposition of the various forms of Zariski s Main Theorem, following EGA. Most of the basic machinery (e.g.

### Resolution of Singularities in Algebraic Varieties

Resolution of Singularities in Algebraic Varieties Emma Whitten Summer 28 Introduction Recall that algebraic geometry is the study of objects which are or locally resemble solution sets of polynomial equations.

### 4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset

4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset Z X. Replacing X by Z we might as well assume that Z

### VIII. Gabber s modification theorem (absolute case) Luc Illusie and Michael Temkin (i) version du à 13h36 TU (19c1b56)

VIII. Gabber s modification theorem (absolute case) Luc Illusie and Michael Temkin (i) version du 2016-11-14 à 13h36 TU (19c1b56) Contents 1. Statement of the main theorem.......................................................

### Minimal Model Theory for Log Surfaces

Publ. RIMS Kyoto Univ. 48 (2012), 339 371 DOI 10.2977/PRIMS/71 Minimal Model Theory for Log Surfaces by Osamu Fujino Abstract We discuss the log minimal model theory for log surfaces. We show that the

### COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski