# DEFINITION OF ABELIAN VARIETIES AND THE THEOREM OF THE CUBE

Size: px
Start display at page:

Transcription

1 DEFINITION OF ABELIAN VARIETIES AND THE THEOREM OF THE CUBE ANGELA ORTEGA (NOTES BY B. BAKKER) Throughout k is a field (not necessarily closed), and all varieties are over k. For a variety X/k, by a basepoint we ll mean a k-rational point x 0 X(k). 1. Definition and examples Definition 1. An abelian variety X/k is a complete variety X with the structure of a group (in the category of varieties). Thus there exists: (1) an identity basepoint e X(k); (2) a multiplication map m : X X X satisfying the associative property; (3) an inverse map i : X X interacting with m in the usual way A homomorphism of abelian varieties is a morphism of varieties respecting the group structure. Remark 2. A more precise definition is that the functor of points of X is given a factorization through the forgetful functor Groups Sets. A homomorphism of abelian varieties is a natural transformation of the corresponding Groups-valued functors of points. Example 3. (1) An elliptic curve E/k is an abelian variety. E can be realized as a plane cubic E P 2, and addition is given by the usual condition that x + y + z = 0 if they are colinear. (2) If Z 2g = Λ C g is a lattice, then the complex torus C g /Λ is a complex (in fact Kähler) manifold with the structure of a group; when it happens to be the C-points of a variety, that variety is an abelian variety. (3) We saw last time that in fact every compact complex group manifold is a torus. (4) If C/k is a curve, the Jacobian Jac(C) is a projective abelian variety, defined over k. In fact, we ll see later that all abelian varieties are projective. (5) For X/C a smooth complete variety, there is a natural map H 1 (X, Z) H 0 (X, Ω 1 X ) given by integration along a cycle, and the albanese is defined as Alb(X) := H 0 (X, Ω 1 X) /H 1 (X, Z) For X = C a complex curve, this is just the Jacobian. Our first aim is to prove an algebraic analog of Example 3.(3) above. Lemma 4. For X, Y, Z varieties with X complete, x 0, y 0, z 0 basepoints, and f : X Y Z a morphism such that f(x y 0 ) = z 0, there is a morphism g so that X Y f Z commutes, where p is projection onto the second factor. p Y g Date: December 11,

2 2 ANGELA ORTEGA (NOTES BY B. BAKKER) Proof. Define g(y) = f(x 0, y). The condition that two morphisms be equal is a closed condition on the source 1, so we need only show f = g p on a nonempty open set. Let z 0 U Z be an open affine, and let F = Z U, G = p(f 1 (F )) Y. X is complete so G Y is the closed set of Y -coordinates that don t arise among the points of X Y that get sent to U. In particular, y 0 Y G =: V which is an open set. Moreover, for all closed points y V, X y gets mapped to U under f by construction. As X is complete, it must be sent to a point, i.e. g(y). Corollary 5. If X, Y are abelian varieties and f : X Y is any morphism, then f(x) = h(x) f(e) for a homomorphism h : X Y. Proof. We may as well assume f(e) = e. Define F : X X Y by F (x, y) = f(xy) f(y) 1 f(x) 1. This sends X e to e, and now apply the lemma. Corollary 6. An abelian variety X is commutative. Proof. Apply Corollary 5 to the inversion map i : X X. From now on, we ll therefore write the group law additively, and denote by 0 the identity. Note that we re really using the completeness of X here; there are many noncommutative connected group schemes (like GL n = Spec k[x][det(x) 1 ]), but they have to be non-complete. 2. Cohomology and base-change This is a really important theorem, and its application to proving the theorem of the cube is as good a time as any to learn it. The seemingly technical heart of the result is the following theorem: Theorem 7. Let f : X Y be a morphism of noetherian schemes with Y = Spec A affine and F a coherent sheaf on X flat over Y. Then there is a finite complex K = [0 K 0 K 1 K n 1 K n 0] of finitely-generated locally free A-modules such that there is an isomorphism of functors H p (X Y Spec B, F A B) = H p (K A B) on the category of A-algebras for each p 0. The key part of this is the finiteness and the finite-generation. Indeed, just taking the Ĉech complex associated to some affine cover of X would give us a complex of flat A-modules universally computing the cohomology, but typically this won t be finite or finitely-generated. Let s see the consequences of this theorem. As a matter of notation, for a point y Y, denote by X(y) = X Y Spec k(y) and F (y) = F OY k(y) the fibers over y. For a scheme X/k we define h p (X, F ) = dim k H p (X, F ), so for example h p (X(y), F (y)) = dim k(y) H p (X(y), F (y)). Corollary 8. In the above situation, for all p 0, (1) y h p (X(y), F (y)) is upper semicontinuous; (2) y χ(x(y), F (y)) is locally constant. No one can ever remember which semicontinuity means which thing, so the above says that cohomology can jump up on special fibers. 1 The set where they re equal is the base-change of the diagonal (and has a natural scheme structure).

3 ABELIAN VARIETIES AND THE THEOREM OF CUBE 3 Proof. Let d p : K p K p+1 be the differential in the complex guaranteed by the theorem. The key idea is that h p (X(y), F (y)) = dim k(y) ker(d p k(y)) dim k(y) im(d p 1 k(y)) = dim k(y) K p k(y) dim k(y) im(d p k(y)) dim k(y) im(d p 1 k(y)) and the first term on the right in the last line is locally constant, while the last two are semicontinuous. Indeed, for any map of sheaves ϕ : E F on Y, the set {y Y rk k(y) (ϕ k(y)) < r} is the zero set of the map r ϕ : r E r F and is closed. In fact, this even gives a natural scheme structure to this set. The euler characteristic is the alternating sum of dim k(y) K p k(y), which is clearly locally constant as the K p are locally free. Corollary 9. Now assume Y is reduced and connected. The following are equivalent: (1) y h p (X(y), F (y)) is constant; (2) R p f F is locally free and the natural map is an isomorphism. (R p f F )(y) = H p (X(y), F (y)) Proof. The backward implication is clear. For the forward direction, we need to know that for E a coherent sheaf on Y, if rk E(y) is constant then it is locally free (this of course uses the reducedness!). By the proof of Corollary 9, if h p (X(y), F (y)) is constant, then dim k(y) im(d p k(y)) and dim k(y) im(d p 1 k(y)) are both constant, which implies im d p, im d p 1, ker d p, and ker d p 1 are all locally free. This gives a splitting of our complex at the pth place: K p 1 K p K p+1 ker d p 1 K p 1 ( 0 ) im d p 1 H K p = ( 0 0 = ) im d p K p+1 Now the theorem says H universally computes the pth cohomology. Right off the bat that means H = R p f F, and base-changing to y, by the canonical maps. H p (X(y), F (y)) = H k(y) = (R p f F )(y) Corollary 10 (Seesaw theorem). For X a complete variety and L a line bundle on X T, the set T 1 = {t T L X t is trivial} is closed in T and L X T1 = p 2 M for a line bundle M on T 1 (with the reduced scheme structure). Proof. First, observe Lemma 11. A line bundle M on a complete variety X is trivial if and only if h 0 (M) > 0 and h 0 (M 1 ) > 0.

4 4 ANGELA ORTEGA (NOTES BY B. BAKKER) Proof. The forward direction is clear. If there is a section O X M and a section O X M 1, then their tensor product is a map O X O X, which must be a constant since neither section was the zero section. But then neither section is ever zero, and hence they are isomorphisms. Now by the lemma, T 1 = {t Y h 0 (L X t ) > 0} {t Y h 0 (L 1 X t ) > 0} which by Corollary 8 is closed. Replace T by T 1 with the reduced scheme structure, so we can assume L is trivial on every fiber of the projection p : X T T. Then by Corollary 9, M = p L is a line bundle. Now, it follows from the fact that the natural map p M L is an isomorphism on every fiber that in fact its an isomorphism. Theorem 12 (Theorem of the cube). Let X, Y be complete varieties, Z any variety, and x 0, y 0, z 0 basepoints. Any line bundle L on X Y Z whose restriction to each of X Y z 0, X y 0 Z, x 0 Y Z is trivial is itself trivial. Let s first give a proof over C using the exponential sequence, at least in the case Z is also compete. For simplicity, assume none of X, Y, Z have torsion in their cohomology, though it won t matter. Then the Künneth theorem tells us that the natural map H (X, Z) H (Y, Z) H (Z, Z) p 1 p 2 p 3 H (X Y Z, Z) (1) is an isomorphism, where p i is the projection to the ith factor (we let p ij be the projection to the i and j factors). Let ι i be the inclusion of the ith factor and ι ij the inclusion of the i and j factors using the basepoints. Concretely, ι 1 : X y 0 z 0 X Y Z, ι 12 : X Y z 0 X Y Z If you think about the isomorphism (1) in degree 2, it means that for any class α H 2 (X Y Z, Z), α = α 12 + α 13 + α 23 α 1 α 2 α 3 where α ij (respectively α i ) is α inserted in the i and jth slots (respectively p i α inserted in the ith slot) via the Künneth formula, so α 12 = ι 12 α 1, α 1 = ι 1α 1 1. In particular, this means if ι ijα = 0 for all i, j, then α = 0 (this is what it means to be quadratic ). The long exact sequence associated to the exponential sequence gives us an exact sequence 0 Z O O 0 H 1 (X Y Z, O) exp H 1 (X Y Z, O ) c 1 H 2 (X Y Z, Z) Given a line bundle L on X Y Z thought of as an element of the middle group, ι ij c 1(L) = c 1 (ι ij L) = 0 for all i, j, so c 1(L) = 0 and L = exp(a) for some A H 1 (X Y Z, O). But H 1 (X, O) H 1 (Y, O) H 1 (Z, O) p 1 +p 2 +p 3 H 1 (X Y Z, O) is an isomorphism, and the hypotheses imply that ι i exp(a) = exp(ι i A) = 0 for each i. Thus, ( ) L = exp ι i A = 0 Here s a sketch of the algebraic proof: i

5 ABELIAN VARIETIES AND THE THEOREM OF CUBE 5 Sketch of proof. By the seesaw theorem, its enough to show that L is trivial when restricted to x Y z for all x z X Z, for then L is a pullback from X Z, but it is trivial on X y 0 Z. We can prove this using the theorem in the case that X is a curve using the following: Lemma 13. For any two x, x X, there is an irreducible curve on X passing through x, x. Proof. This is obvious by Bertini for X projective, but by Chow s lemma this is enough. So now assume X is a curve; after normalizing we can assume X is smooth. Secretly we can conclude because considering L as a family of line bundles on X parametrized by Y Z, we have Y Z Jac(X) but by the hypotheses and Lemma 4, this factors through the projection to Y, i.e. L is a pullback from X Y, and therefore must be trivial, again by the hypotheses. If you don t want to use the existence of the Jacobian, there is a work-around, but I m afraid you ll have to look in Mumford for that. 3. Consequences for abelian varieties First note the following immediate corollary of the theorem of the cube: Corollary 14. For X, Y, Z as in Theorem 12, let L be any line bundle on X Y Z. Then L = p 12L p 13L p 23L p 1L 1 p 2L 1 p 3L 1 Proof. Both sides have the same restriction to X Y z 0 etc. For X an abelian variety, denote by m ij : X X X X the sum of the i and jth coordinates (i.e. m ij = m p ij where m : X X X is the addition map), by m 123 the sum of all three coordinates, and for consistency m i = p i. Corollary 15. Let X be an abelian variety and L a line bundle on X. Then m 123L = m 12L m 13L m 23L m 1L 1 m 2L 1 m 3L 1 Proof. Apply the last corollary to m 123 L. Corollary 16. Let X be any variety, Y an abelian variety, f, g, h : X Y morphisms, and L a line bundle on Y. Then (f + g + h) L = (f + g) L (f + h) L (g + h) L f L 1 g L 1 h L 1 Proof. Pull back the previous corollary along f g h : X Y Y Y. For X an abelian variety, let n : X X be the multiplication by n map, which can be inductively defined by n + 1 = m (n id). We also denote by 1 = i the inversion. The following two results are the most memorable results of this section: Theorem 17. For X an abelian variety and L a line bundle on X, n L = L n2 +n 2 ( 1) L n2 n 2 Proof. The theorem is true for n = 0, 1, 1, and if its true for n its true for n as well. By Corollary 16 applied to f = n, g = 1, h = 1, we have n L = (n + 1) L (n 1) L O n L 1 L 1 ( 1) L 1 Computing this out using the theorem for all the terms except (n + 1) L, we conclude by induction that the result is true for n > 0.

6 6 ANGELA ORTEGA (NOTES BY B. BAKKER) For any k-point x X(k), there is a translation map t x : X X given by t x (y) = y + x. Theorem 18 (Theorem of the square). For any line bundle L on an abelian variety X and any two k-points x, y X(k), t x+yl L = t xl t yl Proof. Apply Corollary 16 to f = id, and g, h the constant maps with images x, y, respectively. This theorem says that for any line bundle L, the map ϕ L : X Pic(X), x t xl L 1 is a homomorphism. Technically at the moment Pic(X) is just the set of line bundles on X defined over k, and the map is only defined set theoretically on the k-points of X, but we ll see soon that this is actually a homomorphism of abelian varieties.

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 43 AND 44

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 43 AND 44 RAVI VAKIL CONTENTS 1. Flat implies constant Euler characteristic 1 2. Proof of Important Theorem on constancy of Euler characteristic in flat families

### 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).

### Algebraic geometry. Contents. Lectures delivered by Xinwen Zhu Notes by Akhil Mathew. Spring 2012, Harvard

Algebraic geometry Lectures delivered by Xinwen Zhu Notes by Akhil Mathew Spring 2012, Harvard Contents Lecture 1 2/6 1 Indefinite integrals 4 2 Abel s work 5 3 Abelian varieties 7 Lecture 2 2/8 1 Abelian

### Fourier Mukai transforms II Orlov s criterion

Fourier Mukai transforms II Orlov s criterion Gregor Bruns 07.01.2015 1 Orlov s criterion In this note we re going to rely heavily on the projection formula, discussed earlier in Rostislav s talk) and

### Math 797W Homework 4

Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

### Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

### 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

### Lecture 4: Abelian varieties (algebraic theory)

Lecture 4: Abelian varieties (algebraic theory) This lecture covers the basic theory of abelian varieties over arbitrary fields. I begin with the basic results such as commutativity and the structure of

### 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

### 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

### Group schemes over fields and Abelian varieties

Group schemes over fields and Abelian varieties Daniele Agostini May 5, 2015 We will work with schemes over a field k, of arbitrary characteristic and not necessarily algebraically closed. An algebraic

### MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23

MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23 6.4. Homotopy uniqueness of projective resolutions. Here I proved that the projective resolution of any R-module (or any object of an abelian category

### 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about

### INTERSECTION THEORY CLASS 19

INTERSECTION THEORY CLASS 19 RAVI VAKIL CONTENTS 1. Recap of Last day 1 1.1. New facts 2 2. Statement of the theorem 3 2.1. GRR for a special case of closed immersions f : X Y = P(N 1) 4 2.2. GRR for closed

### Introduction and preliminaries Wouter Zomervrucht, Februari 26, 2014

Introduction and preliminaries Wouter Zomervrucht, Februari 26, 204. Introduction Theorem. Serre duality). Let k be a field, X a smooth projective scheme over k of relative dimension n, and F a locally

### 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar

### Peter Scholze Notes by Tony Feng. This is proved by real analysis, and the main step is to represent de Rham cohomology classes by harmonic forms.

p-adic Hodge Theory Peter Scholze Notes by Tony Feng 1 Classical Hodge Theory Let X be a compact complex manifold. We discuss three properties of classical Hodge theory. Hodge decomposition. Hodge s theorem

### 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 24

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves

### 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

### SECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS

SECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS In this section we will prove the Künneth theorem which in principle allows us to calculate the (co)homology of product spaces as soon

### 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

### ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY

ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY ARDA H. DEMIRHAN Abstract. We examine the conditions for uniqueness of differentials in the abstract setting of differential geometry. Then we ll come up

### Cohomology and Base Change

Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is half-exact if whenever 0 M M M 0 is an exact sequence of A-modules, the sequence T (M ) T (M)

### 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)

### 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

### MA 206 notes: introduction to resolution of singularities

MA 206 notes: introduction to resolution of singularities Dan Abramovich Brown University March 4, 2018 Abramovich Introduction to resolution of singularities 1 / 31 Resolution of singularities Let k be

### We can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a pre-specified embedding in a projective space. However, an ample line bundle

### 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves of ideals, and closed subschemes 1 2. Invertible sheaves (line bundles) and divisors 2 3. Some line bundles on projective

### Chow Groups. Murre. June 28, 2010

Chow Groups Murre June 28, 2010 1 Murre 1 - Chow Groups Conventions: k is an algebraically closed field, X, Y,... are varieties over k, which are projetive (at worst, quasi-projective), irreducible and

### 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

### p,q H (X), H (Y ) ), where the index p has the same meaning as the

There are two Eilenberg-Moore spectral sequences that we shall consider, one for homology and the other for cohomology. In contrast with the situation for the Serre spectral sequence, for the Eilenberg-Moore

### ABELIAN VARIETIES BRYDEN CAIS

ABELIAN VARIETIES BRYDEN CAIS A canonical reference for the subject is Mumford s book [6], but Mumford generally works over an algebraically closed field (though his arguments can be modified to give results

### Coherent sheaves on elliptic curves.

Coherent sheaves on elliptic curves. Aleksei Pakharev April 5, 2017 Abstract We describe the abelian category of coherent sheaves on an elliptic curve, and construct an action of a central extension of

### 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.

### Theta divisors and the Frobenius morphism

Theta divisors and the Frobenius morphism David A. Madore Abstract We introduce theta divisors for vector bundles and relate them to the ordinariness of curves in characteristic p > 0. We prove, following

### INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 23

INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 23 RAVI VAKIL Contents 1. More background on invertible sheaves 1 1.1. Operations on invertible sheaves 1 1.2. Maps to projective space correspond to a vector

### Porteous s Formula for Maps between Coherent Sheaves

Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for

### Curves on P 1 P 1. Peter Bruin 16 November 2005

Curves on P 1 P 1 Peter Bruin 16 November 2005 1. Introduction One of the exercises in last semester s Algebraic Geometry course went as follows: Exercise. Let be a field and Z = P 1 P 1. Show that the

### 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.

### 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

### ne varieties (continued)

Chapter 2 A ne varieties (continued) 2.1 Products For some problems its not very natural to restrict to irreducible varieties. So we broaden the previous story. Given an a ne algebraic set X A n k, we

### INTERSECTION THEORY CLASS 6

INTERSECTION THEORY CLASS 6 RAVI VAKIL CONTENTS 1. Divisors 2 1.1. Crash course in Cartier divisors and invertible sheaves (aka line bundles) 3 1.2. Pseudo-divisors 3 2. Intersecting with divisors 4 2.1.

### Locally G-ringed spaces and rigid spaces

18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 Rigid analytic spaces (at last!) We are now ready to talk about rigid analytic spaces in earnest. I ll give the

### Applications to the Beilinson-Bloch Conjecture

Applications to the Beilinson-Bloch Conjecture Green June 30, 2010 1 Green 1 - Applications to the Beilinson-Bloch Conjecture California is like Italy without the art. - Oscar Wilde Let X be a smooth projective

### Math 6510 Homework 10

2.2 Problems 9 Problem. Compute the homology group of the following 2-complexes X: a) The quotient of S 2 obtained by identifying north and south poles to a point b) S 1 (S 1 S 1 ) c) The space obtained

### 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

### NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY

NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc

### The Riemann-Roch Theorem

The Riemann-Roch Theorem TIFR Mumbai, India Paul Baum Penn State 7 August, 2015 Five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. Beyond ellipticity 5. The Riemann-Roch

### MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA

MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to

### 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

### NORMALIZATION OF THE KRICHEVER DATA. Motohico Mulase

NORMALIZATION OF THE KRICHEVER DATA Motohico Mulase Institute of Theoretical Dynamics University of California Davis, CA 95616, U. S. A. and Max-Planck-Institut für Mathematik Gottfried-Claren-Strasse

### 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

### ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0.

ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0. ANDREW SALCH During the last lecture, we found that it is natural (even just for doing undergraduatelevel complex analysis!)

### Isogeny invariance of the BSD conjecture

Isogeny invariance of the BSD conjecture Akshay Venkatesh October 30, 2015 1 Examples The BSD conjecture predicts that for an elliptic curve E over Q with E(Q) of rank r 0, where L (r) (1, E) r! = ( p

### Tensor, Tor, UCF, and Kunneth

Tensor, Tor, UCF, and Kunneth Mark Blumstein 1 Introduction I d like to collect the basic definitions of tensor product of modules, the Tor functor, and present some examples from homological algebra and

### A Primer on Homological Algebra

A Primer on Homological Algebra Henry Y Chan July 12, 213 1 Modules For people who have taken the algebra sequence, you can pretty much skip the first section Before telling you what a module is, you probably

### Here is another way to understand what a scheme is 1.GivenaschemeX, and a commutative ring R, the set of R-valued points

Chapter 7 Schemes III 7.1 Functor of points Here is another way to understand what a scheme is 1.GivenaschemeX, and a commutative ring R, the set of R-valued points X(R) =Hom Schemes (Spec R, X) This is

### Geometric Class Field Theory

Geometric Class Field Theory Notes by Tony Feng for a talk by Bhargav Bhatt April 4, 2016 In the first half we will explain the unramified picture from the geometric point of view, and in the second half

### Overview of Atiyah-Singer Index Theory

Overview of Atiyah-Singer Index Theory Nikolai Nowaczyk December 4, 2014 Abstract. The aim of this text is to give an overview of the Index Theorems by Atiyah and Singer. Our primary motivation is to understand

### The Canonical Sheaf. Stefano Filipazzi. September 14, 2015

The Canonical Sheaf Stefano Filipazzi September 14, 015 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will go over

### 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

### Proof of Langlands for GL(2), II

Proof of Langlands for GL(), II Notes by Tony Feng for a talk by Jochen Heinloth April 8, 016 1 Overview Let X/F q be a smooth, projective, geometrically connected curve. The aim is to show that if E is

### Intermediate Jacobians and Abel-Jacobi Maps

Intermediate Jacobians and Abel-Jacobi Maps Patrick Walls April 28, 2012 Introduction Let X be a smooth projective complex variety. Introduction Let X be a smooth projective complex variety. Intermediate

### INTERSECTION THEORY CLASS 7

INTERSECTION THEORY CLASS 7 RAVI VAKIL CONTENTS 1. Intersecting with a pseudodivisor 1 2. The first Chern class of a line bundle 3 3. Gysin pullback 4 4. Towards the proof of the big theorem 4 4.1. Crash

### 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

### 1 Existence of the Néron model

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

### Hecke modifications. Aron Heleodoro. May 28, 2013

Hecke modifications Aron Heleodoro May 28, 2013 1 Introduction The interest on Hecke modifications in the geometrical Langlands program comes as a natural categorification of the product in the spherical

### Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013 As usual, a curve is a smooth projective (geometrically irreducible) variety of dimension one and k is a perfect field. 23.1

### APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY

APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY In this appendix we begin with a brief review of some basic facts about singular homology and cohomology. For details and proofs, we refer to [Mun84]. We then

### 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

### VERY ROUGH NOTES ON SPECTRAL DEFORMATION THEORY

VERY ROUGH NOTES ON SPECTRAL DEFORMATION THEORY 1. Classical Deformation Theory I want to begin with some classical deformation theory, before moving on to the spectral generalizations that constitute

### 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Application of cohomology: Hilbert polynomials and functions, Riemann- Roch, degrees, and arithmetic genus 1 1. APPLICATION OF COHOMOLOGY:

### Deligne s Mixed Hodge Structure for Projective Varieties with only Normal Crossing Singularities

Deligne s Mixed Hodge Structure for Projective Varieties with only Normal Crossing Singularities B.F Jones April 13, 2005 Abstract Following the survey article by Griffiths and Schmid, I ll talk about

### On Mordell-Lang in Algebraic Groups of Unipotent Rank 1

On Mordell-Lang in Algebraic Groups of Unipotent Rank 1 Paul Vojta University of California, Berkeley and ICERM (work in progress) Abstract. In the previous ICERM workshop, Tom Scanlon raised the question

### APPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP

APPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP In this appendix we review some basic facts about étale cohomology, give the definition of the (cohomological) Brauer group, and discuss

### PERVERSE SHEAVES: PART I

PERVERSE SHEAVES: PART I Let X be an algebraic variety (not necessarily smooth). Let D b (X) be the bounded derived category of Mod(C X ), the category of left C X -Modules, which is in turn a full subcategory

### 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

### BRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,

CONNECTIONS, CURVATURE, AND p-curvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and p-curvature, in terms of maps of sheaves

### 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.

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5 RAVI VAKIL CONTENTS 1. The inverse image sheaf 1 2. Recovering sheaves from a sheaf on a base 3 3. Toward schemes 5 4. The underlying set of affine schemes 6 Last

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Where we were 1 2. Yoneda s lemma 2 3. Limits and colimits 6 4. Adjoints 8 First, some bureaucratic details. We will move to 380-F for Monday

### 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

### 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

### 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

### Beilinson s conjectures I

Beilinson s conjectures I Akshay Venkatesh February 17, 2016 1 Deligne s conjecture As we saw, Deligne made a conjecture for varieties (actually at the level of motives) for the special values of L-function.

### 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:

### An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations

An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations Mircea Mustaţă University of Michigan Mainz July 9, 2018 Mircea Mustaţă () An overview of D-modules Mainz July 9, 2018 1 The

### Construction of M B, M Dol, M DR

Construction of M B, M Dol, M DR Hendrik Orem Talbot Workshop, Spring 2011 Contents 1 Some Moduli Space Theory 1 1.1 Moduli of Sheaves: Semistability and Boundedness.............. 1 1.2 Geometric Invariant

### Realizing Families of Landweber Exact Theories

Realizing Families of Landweber Exact Theories Paul Goerss Department of Mathematics Northwestern University Summary The purpose of this talk is to give a precise statement of 1 The Hopkins-Miller Theorem

### Algebraic Geometry

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

### Hodge Theory of Maps

Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar

### PERVERSE SHEAVES. Contents

PERVERSE SHEAVES SIDDHARTH VENKATESH Abstract. These are notes for a talk given in the MIT Graduate Seminar on D-modules and Perverse Sheaves in Fall 2015. In this talk, I define perverse sheaves on a

### 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