# h M (T ). The natural isomorphism η : M h M determines an element U = η 1

Size: px
Start display at page:

Download "h M (T ). The natural isomorphism η : M h M determines an element U = η 1"

Transcription

1 MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition Let M : Sch Set be a moduli functor; then a scheme M is a fine moduli space for M if it represents M. Let s carefully unravel this definition: M is a fine moduli space for M if there is a natural isomorphism η : M h M. Hence, for every scheme S, we have a bijection η S : M(S) := {families over S}/ S h M (S) := {morphisms S M}. In particular, if S = Spec k, then the k-points of M are in bijection with the set A/. Furthermore, these bijections are compatible with morphisms T S, in the sense that we have a commutative diagram η S M(S) h M (S) M(f) h M (f) M(T ) η T h M (T ). The natural isomorphism η : M h M determines an element U = η 1 M (id M) M(M); that is, U is a family over M (up to equivalence). Definition Let M be a fine moduli space for M; then the family U M(M) corresponding to the identity morphism on M is called the universal family. This family is called the universal family, as any family F over a scheme S (up to equivalence) corresponds to a morphism f : S M and, moreover, as the families f U and F correspond to the same morphism id M f = f, we have f U S F; that is, any family is equivalent to a family obtained by pulling back the universal family. Remark If a fine moduli space for M exists, it is unique up to unique isomorphism: that is, if (M, η) and (M, η ) are two fine moduli spaces, then they are related by unique isomorphisms η M ((η M) 1 (Id M )) : M M and η M ((η M ) 1 (Id M )) : M M. We recall that a presheaf F : Sch Set is said to be a sheaf in the Zariski topology if for every scheme S and Zariski cover {S i } of S, the natural map {f F (S)} {(f i F (S i )) i : f i Si S j = f j Sj S i for all i, j} is a bijection. A presheaf is called a separated presheaf if these natural maps are injective. Exercise (1) Show that the functor of points of a scheme is a sheaf in the Zariski topology. In particular, deduce that for a presheaf to be representable it must be a sheaf in the Zariski topology. (2) Consider the moduli functor of vector bundles over a fixed scheme X, where we say two families E and F are equivalent if and only if they are isomorphic. Show that the corresponding moduli functor fails to be a separable presheaf (it may be useful to consider the second equivalence relation we introduced for families of vector bundles in Exercise 2.12). Example Let us consider the projective space P n = Proj k[x 0,..., x n ]. This variety can be interpreted as a fine moduli space for the moduli problem of lines through the origin in V := A n+1. To define this moduli problem carefully, we need to define a notion of families and equivalences of families. A family of lines through the origin in V over a scheme S is a line bundle L over S which is a subbundle of the trivial vector bundle V S over S (by subbundle we mean that the quotient is also a vector bundle). Then two families are equivalent if and only if they are equal.

2 8 VICTORIA HOSKINS Over P n, we have a tautological line bundle O P n( 1) V P n, whose fibre over p P n is the corresponding line in V. This provides a tautological family of lines over P n. The dual of the tautological line bundle is the line bundle O P n(1), known as the Serre twisting sheaf. The important fact we need about O P n(1) is that it is generated by the global sections x 0,..., x n. Given any morphism of schemes f : S P n, the line bundle f O P n(1) is generated by the global sections f (x 0 ),..., f (x n ). Hence, we have a surjection O n+1 S f O P n(1). For locally free sheaves, pull back commutes with dualising and so f O P n( 1) = (f O P n(1)). Dually the above surjection gives an inclusion L := f O P n( 1) O n+1 S = V S which determines a family of lines in V over S. Conversely, let L V S be a family of lines through the origin in V over S. Then, dual to this inclusion, we have a surjection q : V S L. The vector bundle V S is generated by the global sections σ 0,..., σ n corresponding to the dual basis for the standard basis on V. Since q is surjective, the dual line bundle L is generated by the global sections q σ 0,..., q σ n. In particular, there is a unique morphism f : S P n given by s [q σ 0 (s) : : q σ n (s)] such that f O P n( 1) = L V S (for details, see [14] II Theorem 7.1). Hence, there is a bijective correspondence between morphisms S P n and families of lines through the origin in V over S. In particular, P n is a fine moduli space and the tautological family is a universal family. The keen reader may note that the above calculations suggests we should rather think of P n as the space of 1-dimensional quotient spaces of a n + 1-dimensional vector space (a convention that many algebraic geometers use). Exercise Consider the moduli problem of d-dimensional linear subspaces in a fixed vector space V = A n, where a family over S is a rank d vector subbundle E of V S and the equivalence relation is given by equality. We denote the associated moduli functor by Gr(d, n). We recall that there is a projective variety Gr(d, n) whose k-points parametrise d-dimensional linear subspaces of k n, called the Grassmannian variety. Let T V Gr(d, n) be the tautological family over Gr(d, n) whose fibre over a point in the Grassmannian is the corresponding linear subspace of V. In this exercise, we will show that the Grassmannian variety Gr(d, n) is a fine moduli space representing Gr(d, n). Let us determine the natural isomorphism η : Gr(d, n) h Gr(d,n). Consider a family E V S over S. As E is a rank d vector bundle, we can pick an open cover {U α } of S on which E is trivial, i.e. E Uα = Uα A d. Then, since we have U α A d = E Uα V S Uα = A n U α, we obtain a homomorphism U α A d U α A n of trivial vector bundles over U α. This determines a n d matrix with coefficients in O(U α ) of rank d; that is, a morphism U α Mn d d (k), to the variety of n d matrices of rank d. By taking the wedge product of the d rows in this matrix, we obtain a morphism f α : U α P( d (k n )) with image in the Grassmannian Gr(d, n). Using the fact that the transition functions of E are linear, verify that these morphisms glue to define a morphism f = f E : S P( d (k n )) such that f T = E. In particular, this procedure determines the natural isomorphism: η S (E) = f E. For a comprehensive coverage of the Grassmannian moduli functor and its representability, see [8] Section 8. The Grassmannian moduli functor has a natural generalisation to the moduli problem of classifying subsheaves of a fixed sheaf (or equivalently quotient sheaves with a natural notion of equivalence). This functor is representable by a quot scheme constructed by Grothendieck [9, 10] (for a survey of the construction, see [33]). Let us mention two special cases of this construction. Firstly, if we take our fixed sheaf to be the structure sheaf of a scheme X, then we are considering ideal sheaves and obtain a Hilbert scheme classifying subschemes of X. Secondly, if we take our fixed sheaf to be a locally free coherent sheaf E over X and consider quotient line bundles of E, we obtain the projective space bundle P(E) over X (see [14] II 7) Pathological behaviour. Unfortunately, there are many natural moduli problems which do not admit a fine moduli space. In this section, we study some examples and highlight two

3 MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 9 particular pathologies which prevent a moduli problem from admitting a fine moduli space, namely: (1) The jump phenomena: moduli may jump in families (in the sense that we can have a family F over A 1 such that F s F s for all s, s A 1 {0}, but F 0 F s for s A 1 {0}). (2) The moduli problem may be unbounded (in that there is no family F over a scheme S which parametrises all objects in the moduli problem). Example We consider the naive moduli problem of classifying endomorphisms of a n-dimensional k-vector space. More precisely A consists of pairs (V, T ), where V is an n- dimensional k-vector space and T is an endomorphism of V. We say (V, φ) (V, φ ) if there exists an isomorphism h : V V compatible with the endomorphisms i.e. h φ = φ h. We extend this to a moduli problem by defining a family over S to be a rank n vector bundle F over S with an endomorphism φ : F F. Then we say (F, φ) S (G, φ ) if there is an isomorphism h : F G such that h φ = φ h. Let End n be the corresponding moduli functor. For any n 2, we can construct families which exhibit the jump phenomena. For concreteness, let n = 2. Then consider the family over A 1 given by (F = O 2, φ) where for s A 1 A 1 φ s = ( 1 s 0 1 For s, t 0, these matrices are similar and so φ t φ s. However, φ 0 φ 1, as this matrices have distinct Jordan normal forms. Hence, we have produced a family with the jump phenomenon. Example Let us consider the moduli problem of vector bundles over P 1 of rank 2 and degree 0. We claim there is no family F over a scheme S with the property that for any rank 2 degree 0 vector bundle E on P 1, there is a k-point s S such that F s = E. Suppose such a family F over a scheme S exists. For each n N, we have a rank 2 degree 0 vector bundle O P 1(n) O P 1( n) (in fact, by Grothendieck s Theorem classifying vector bundles on P 1, every rank 2 degree 0 vector bundle on P 1 has this form). Furthermore, we have { dim H 0 (P 1 2 if n = 0,, O P 1(n) O P 1( n)) = dim k (k[x 0, x 1 ] n k[x 0, x 1 ] n ) = n + 1 if n 1. Consider the subschemes S n := {s S : dim H 0 (P 1, F s ) n} of S, which are closed by the semi-continuity theorem (see [14] III Theorem 12.8). Then we obtain a decreasing chain of closed subschemes S = S 2 S 3 S 4... each of which is distinct as O P 1(n) O P 1( n) S n+1 S n+2. The existence of this chain contradicts the fact that S is Noetherian (recall that for us scheme means scheme of finite type over k). In particular, the moduli problem of vector bundles of rank 2 and degree 0 is unbounded. In fact, we also see the jump phenomena: there is a family F of rank 2 degree 0 vector bundles over A 1 = Spec k[s] such that { O 2 s 0 F s = P 1 O P 1(1) O P 1( 1) s = 0. To construct this family, we note that Ext 1 (O P 1(1), O P 1( 1)) = H 1 (P 1, O P 1( 2)) = H 0 (P 1, O P 1) = k by Serre duality. Therefore, there is a family of extensions F over A 1 of O P 1(1) by O P 1( 1) with the desired property. In both cases there is no fine moduli space for this problem. To solve these types of phenomena, one usually restricts to a nicer class of objects (we will return to this idea later on). ).

4 10 VICTORIA HOSKINS Example We can see more directly that there is no fine moduli space for End n. Suppose M is a fine moduli space. Then we have a bijection between morphisms S M and families over S up to equivalence. Choose any n n matrix T, which determines a point m M. Then for S = P 1 we have that the trivial families (O n P 1, T ) and (O n P 1 O P 1(1), T Id OP 1 (1)) are non-equivalent families which determine the same morphism P 1 M, namely the constant morphism to the point m Coarse moduli spaces. As demonstrated by the above examples, not every moduli functor has a fine moduli space. By only asking for a natural transformation M h M which is universal and a bijection over Spec k (so that the k-points of M are in bijection with the equivalence classes A/ ), we obtain a weaker notion of a coarse moduli space. Definition A coarse moduli space for a moduli functor M is a scheme M and a natural transformation of functors η : M h M such that (a) η Spec k : M(Spec k) h M (Spec k) is bijective. (b) For any scheme N and natural transformation ν : M h N, there exists a unique morphism of schemes f : M N such that ν = h f η, where h f : h M h N is the corresponding natural transformation of presheaves. Remark A coarse moduli space for M is unique up to unique isomorphism: if (M, η) and (M, η ) are coarse moduli spaces for M, then by Property (b) there exists unique morphisms f : M M and f : M M such that h f and h f fit into two commutative triangles: h M η M η h M h f η η h f h M h M. Since η = h f h f η and η = h idm η, by uniqueness in (b) and the Yoneda Lemma, we have f f = id M and similarly f f = id M. Proposition Let (M, η) be a coarse moduli space for a moduli problem M. Then (M, η) is a fine moduli space if and only if (1) there exists a family U over M such that η M (U) = id M, (2) for families F and G over a scheme S, we have F S G η S (F) = η S (G). Proof. Exercise. Lemma Let M be a moduli problem and suppose there exists a family F over A 1 such that F s F 1 for all s 0 and F 0 F 1. Then for any scheme M and natural transformation η : M h M, we have that η A 1(F) : A 1 M is constant. In particular, there is no coarse moduli space for this moduli problem. Proof. Suppose we have a natural transformation η : M h M ; then η sends the family F over A 1 to a morphism f : A 1 M. For any s : Spec k A 1, we have that f s = η Spec k (F s ) and, for s 0, F s = F 1 M(Spec k), so that f A 1 {0} is a constant map. Let m : Spec k M be the point corresponding to the equivalence class for F 1 under η. Since the k-valued points of M are closed (recall M is a scheme of finite type over an algebraically closed field), their preimages under morphisms must also be closed. Then, as A 1 {0} f 1 (m), the closure A 1 of A 1 {0} must also be contained in f 1 (m); that is, f is the constant map to the k-valued point m of M. In particular, the map η Spec k : M(Spec k) h M (Spec k) is not a bijection, as F 0 F 1 in M(Spec k), but these non-equivalent objects correspond to the same k-point m in M. In particular, the moduli problems of Examples 2.22 and 2.21 do not even admit coarse moduli spaces.

5 MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY The construction of moduli spaces. The construction of many moduli spaces follows the same general pattern. (1) Fix any discrete invariants for our objects - here the invariants should be invariant under the given equivalence relation (for example, for isomorphism classes of vector bundles on a curve, one may fix the rank and degree). (2) Restrict to a reasonable class of objects which are bounded (otherwise, we can t find a coarse moduli space). Usually one restricts to a class of stable objects which are better behaved and bounded. (3) Find a family F over a scheme P with the local universal property (i.e. locally any other family is equivalent to a pullback of this family - see below). We call P a parameter space, as the k-points of P surject onto A/ ; however, this is typically not a bijection. (4) Find a group G acting on P such that p and q lie in the same G-orbit in P if and only if F p F q. Then we have a bijection P (k)/g = A/. (5) Typically this group action is algebraic (see Section 3) and by taking a quotient, we should obtain our moduli space. The quotient should be taken in the category of schemes (in terminology to come, it should be a categorical quotient) and this is done using Mumford s Geometric Invariant Theory. Definition For a moduli problem M, a family F over a scheme S has the local universal property if for any other family G over a scheme T and for any k-point t T, there exists a neighbourhood U of t in T and a morphism f : U S such that G U U f F. In particular, we do not require the morphism f to be unique. We note that, for such a family to exist, we need our moduli problem to be bounded.

### MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY VICTORIA HOSKINS Abstract In this course, we study moduli problems in algebraic geometry and the construction of moduli spaces using geometric invariant theory.

### 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

### What are stacks and why should you care?

What are stacks and why should you care? Milan Lopuhaä October 12, 2017 Todays goal is twofold: I want to tell you why you would want to study stacks in the first place, and I want to define what a stack

### 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.

### Moduli Theory and Geometric Invariant Theory

Moduli Theory and Geometric Invariant Theory November 29, 2017 Chapter 1 Moduli theory 1.1 Category theory A category consists of a collection of objects, and for each pair of objects, a set of morphisms

### 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

### 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 moduli stack of vector bundles on a curve

The moduli stack of vector bundles on a curve Norbert Hoffmann norbert.hoffmann@fu-berlin.de Abstract This expository text tries to explain briefly and not too technically the notions of stack and algebraic

### PICARD GROUPS OF MODULI PROBLEMS II

PICARD GROUPS OF MODULI PROBLEMS II DANIEL LI 1. Recap Let s briefly recall what we did last time. I discussed the stack BG m, as classifying line bundles by analyzing the sense in which line bundles may

### 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

### THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3

THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3 KAZUNORI NAKAMOTO AND TAKESHI TORII Abstract. There exist 26 equivalence classes of k-subalgebras of M 3 (k) for any algebraically closed field

### 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 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)

### 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

### 14 Lecture 14: Basic generallities on adic spaces

14 Lecture 14: Basic generallities on adic spaces 14.1 Introduction The aim of this lecture and the next two is to address general adic spaces and their connection to rigid geometry. 14.2 Two open questions

### Representations and Linear Actions

Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category

### 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

### ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset

### Direct Limits. Mathematics 683, Fall 2013

Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry

### CATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS. Contents. 1. The ring K(R) and the group Pic(R)

CATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS J. P. MAY Contents 1. The ring K(R) and the group Pic(R) 1 2. Symmetric monoidal categories, K(C), and Pic(C) 2 3. The unit endomorphism ring R(C ) 5 4.

### AN INTRODUCTION TO MODULI SPACES OF CURVES CONTENTS

AN INTRODUCTION TO MODULI SPACES OF CURVES MAARTEN HOEVE ABSTRACT. Notes for a talk in the seminar on modular forms and moduli spaces in Leiden on October 24, 2007. CONTENTS 1. Introduction 1 1.1. References

### AN INTRODUCTION TO AFFINE SCHEMES

AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geometry. The goal of this paper is to present the basic concepts of algebraic geometry,

### MODULI TOPOLOGY. 1. Grothendieck Topology

MODULI TOPOLOG Abstract. Notes from a seminar based on the section 3 of the paper: Picard groups of moduli problems (by Mumford). 1. Grothendieck Topology We can define a topology on any set S provided

### 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

### ABSTRACT NONSINGULAR CURVES

ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine n-space the collection A n k of points P = a 1, a,..., a

### Twisted rings and moduli stacks of fat point modules in non-commutative projective geometry

Twisted rings and moduli stacks of fat point modules in non-commutative projective geometry DANIEL CHAN 1 University of New South Wales e-mail address:danielc@unsw.edu.au Abstract The Hilbert scheme of

### GEOMETRIC INVARIANT THEORY AND SYMPLECTIC QUOTIENTS

GEOMETRIC INVARIANT THEORY AND SYMPLECTIC QUOTIENTS VICTORIA HOSKINS 1. Introduction In this course we study methods for constructing quotients of group actions in algebraic and symplectic geometry and

### 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

### 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).

### Lecture 9: Sheaves. February 11, 2018

Lecture 9: Sheaves February 11, 2018 Recall that a category X is a topos if there exists an equivalence X Shv(C), where C is a small category (which can be assumed to admit finite limits) equipped with

### 3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection

3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection is called the objects of C and is denoted Obj(C). Given

### 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

### 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

### Section Higher Direct Images of Sheaves

Section 3.8 - Higher Direct Images of Sheaves Daniel Murfet October 5, 2006 In this note we study the higher direct image functors R i f ( ) and the higher coinverse image functors R i f! ( ) which will

### 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.

### 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

### Solutions to some of the exercises from Tennison s Sheaf Theory

Solutions to some of the exercises from Tennison s Sheaf Theory Pieter Belmans June 19, 2011 Contents 1 Exercises at the end of Chapter 1 1 2 Exercises in Chapter 2 6 3 Exercises at the end of Chapter

### Algebraic Geometry I Lectures 14 and 15

Algebraic Geometry I Lectures 14 and 15 October 22, 2008 Recall from the last lecture the following correspondences {points on an affine variety Y } {maximal ideals of A(Y )} SpecA A P Z(a) maximal ideal

### 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

### 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

### 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

### Systems of linear equations. We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K,

Systems of linear equations We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K, f 11 t 1 +... + f 1n t n = 0, f 21 t 1 +... + f 2n t n = 0,.

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 26

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 26 RAVI VAKIL CONTENTS 1. Proper morphisms 1 Last day: separatedness, definition of variety. Today: proper morphisms. I said a little more about separatedness of

### 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

### 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

### Elementary (ha-ha) Aspects of Topos Theory

Elementary (ha-ha) Aspects of Topos Theory Matt Booth June 3, 2016 Contents 1 Sheaves on topological spaces 1 1.1 Presheaves on spaces......................... 1 1.2 Digression on pointless topology..................

### PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES

PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES Angelo Vistoli Scuola Normale Superiore Bordeaux, June 23, 2010 Joint work with Niels Borne Université de Lille 1 Let X be an algebraic variety over C, x 0 X. What

### FORMAL GLUEING OF MODULE CATEGORIES

FORMAL GLUEING OF MODULE CATEGORIES BHARGAV BHATT Fix a noetherian scheme X, and a closed subscheme Z with complement U. Our goal is to explain a result of Artin that describes how coherent sheaves on

### The Moduli Space of Rank 2 Vector Bundles on Projective Curves

The Moduli Space of Rank Vector Bundles on Projective urves T. Nakamura, A. Simonetti 1 Introduction Let be a smooth projective curve over and let L be a line bundle on. onsider the moduli functor VB (,

### The derived category of a GIT quotient

September 28, 2012 Table of contents 1 Geometric invariant theory 2 3 What is geometric invariant theory (GIT)? Let a reductive group G act on a smooth quasiprojective (preferably projective-over-affine)

### ON TAME STACKS IN POSITIVE CHARACTERISTIC

ON TAME STACKS IN POSITIVE CHARACTERISTIC DAN ABRAMOVICH, MARTIN OLSSON, AND ANGELO VISTOLI Contents 1. Linearly reductive finite group schemes 1 2. Tame stacks 13 3. Twisted stable maps 20 4. Reduction

### 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)

### 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

### PART II.2. THE!-PULLBACK AND BASE CHANGE

PART II.2. THE!-PULLBACK AND BASE CHANGE Contents Introduction 1 1. Factorizations of morphisms of DG schemes 2 1.1. Colimits of closed embeddings 2 1.2. The closure 4 1.3. Transitivity of closure 5 2.

### AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES

AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES MATTHEW H. BAKER AND JÁNOS A. CSIRIK This paper was written in conjunction with R. Hartshorne s Spring 1996 Algebraic Geometry course at

### Algebraic varieties. Chapter A ne varieties

Chapter 4 Algebraic varieties 4.1 A ne varieties Let k be a field. A ne n-space A n = A n k = kn. It s coordinate ring is simply the ring R = k[x 1,...,x n ]. Any polynomial can be evaluated at a point

### 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

### VECTOR BUNDLES ON THE PROJECTIVE LINE AND FINITE DOMINATION OF CHAIN COMPLEXES

VECTOR BUNDLES ON THE PROJECTIVE LINE AND FINITE DOMINATION OF CHAIN COMPLEXES THOMAS HÜTTEMANN Abstract. We present an algebro-geometric approach to a theorem on finite domination of chain complexes over

### 0.1 Spec of a monoid

These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.

### 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

### Azumaya Algebras. Dennis Presotto. November 4, Introduction: Central Simple Algebras

Azumaya Algebras Dennis Presotto November 4, 2015 1 Introduction: Central Simple Algebras Azumaya algebras are introduced as generalized or global versions of central simple algebras. So the first part

### PULLBACK MODULI SPACES

PULLBACK MODULI SPACES FRAUKE M. BLEHER AND TED CHINBURG Abstract. Geometric invariant theory can be used to construct moduli spaces associated to representations of finite dimensional algebras. One difficulty

### Algebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra

Algebraic Varieties Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra Algebraic varieties represent solutions of a system of polynomial

### Lecture 2 Sheaves and Functors

Lecture 2 Sheaves and Functors In this lecture we will introduce the basic concept of sheaf and we also will recall some of category theory. 1 Sheaves and locally ringed spaces The definition of sheaf

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETR CLASS 24 RAVI VAKIL CONTENTS 1. Normalization, continued 1 2. Sheaf Spec 3 3. Sheaf Proj 4 Last day: Fibers of morphisms. Properties preserved by base change: open immersions,

### IndCoh Seminar: Ind-coherent sheaves I

IndCoh Seminar: Ind-coherent sheaves I Justin Campbell March 11, 2016 1 Finiteness conditions 1.1 Fix a cocomplete category C (as usual category means -category ). This section contains a discussion of

### LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES

LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EAMPLES VIVEK SHENDE Historically, sheaves come from topology and analysis; subsequently they have played a fundamental role in algebraic geometry and certain

### 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

### 3. The Sheaf of Regular Functions

24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

### Homology and Cohomology of Stacks (Lecture 7)

Homology and Cohomology of Stacks (Lecture 7) February 19, 2014 In this course, we will need to discuss the l-adic homology and cohomology of algebro-geometric objects of a more general nature than algebraic

### Topological K-theory, Lecture 3

Topological K-theory, Lecture 3 Matan Prasma March 2, 2015 1 Applications of the classification theorem continued Let us see how the classification theorem can further be used. Example 1. The bundle γ

### 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!)

### 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

### Lectures on Galois Theory. Some steps of generalizations

= Introduction Lectures on Galois Theory. Some steps of generalizations Journée Galois UNICAMP 2011bis, ter Ubatuba?=== Content: Introduction I want to present you Galois theory in the more general frame

### ALGEBRAIC GROUPS JEROEN SIJSLING

ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined

### Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle

Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 7-9 December 2010 1 The notion of vector bundle In affine geometry,

### 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

### D-manifolds and derived differential geometry

D-manifolds and derived differential geometry Dominic Joyce, Oxford University September 2014 Based on survey paper: arxiv:1206.4207, 44 pages and preliminary version of book which may be downloaded from

### Math 535a Homework 5

Math 535a Homework 5 Due Monday, March 20, 2017 by 5 pm Please remember to write down your name on your assignment. 1. Let (E, π E ) and (F, π F ) be (smooth) vector bundles over a common base M. A vector

### 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

### Mini-Course on Moduli Spaces

Mini-Course on Moduli Spaces Emily Clader June 2011 1 What is a Moduli Space? 1.1 What should a moduli space do? Suppose that we want to classify some kind of object, for example: Curves of genus g, One-dimensional

### 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

### ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES N. MOHAN KUMAR, A. P. RAO, AND G. V. RAVINDRA Abstract. We prove that any rank two arithmetically Cohen- Macaulay vector bundle on a general hypersurface

### Boolean Algebras, Boolean Rings and Stone s Representation Theorem

Boolean Algebras, Boolean Rings and Stone s Representation Theorem Hongtaek Jung December 27, 2017 Abstract This is a part of a supplementary note for a Logic and Set Theory course. The main goal is to

### QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS

QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS SAM RASKIN 1. Differential operators on stacks 1.1. We will define a D-module of differential operators on a smooth stack and construct a symbol map when

### LECTURE 3: RELATIVE SINGULAR HOMOLOGY

LECTURE 3: RELATIVE SINGULAR HOMOLOGY In this lecture we want to cover some basic concepts from homological algebra. These prove to be very helpful in our discussion of singular homology. The following

### Finite group schemes

Finite group schemes Johan M. Commelin October 27, 2014 Contents 1 References 1 2 Examples 2 2.1 Examples we have seen before.................... 2 2.2 Constant group schemes....................... 3 2.3

### Vertex algebras, chiral algebras, and factorisation algebras

Vertex algebras, chiral algebras, and factorisation algebras Emily Cliff University of Illinois at Urbana Champaign 18 September, 2017 Section 1 Vertex algebras, motivation, and road-plan Definition A

### A QUICK NOTE ON ÉTALE STACKS

A QUICK NOTE ON ÉTALE STACKS DAVID CARCHEDI Abstract. These notes start by closely following a talk I gave at the Higher Structures Along the Lower Rhine workshop in Bonn, in January. I then give a taste

### 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

### 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.

### Vector Bundles vs. Jesko Hüttenhain. Spring Abstract

Vector Bundles vs. Locally Free Sheaves Jesko Hüttenhain Spring 2013 Abstract Algebraic geometers usually switch effortlessly between the notion of a vector bundle and a locally free sheaf. I will define

### BEZOUT S THEOREM CHRISTIAN KLEVDAL

BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this

### 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

### Notes on Partial Resolutions of Nilpotent Varieties by Borho and Macpherson

Notes on Partial Resolutions of Nilpotent Varieties by Borho and Macpherson Chris Elliott January 14th, 2014 1 Setup Let G be a complex reductive Lie group with Lie algebra g. The paper [BM83] relates

### 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,

### 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,

### 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