A ne Algebraic Varieties Undergraduate Seminars: Toric Varieties
|
|
- Bryan Marsh
- 5 years ago
- Views:
Transcription
1 A ne Algebraic Varieties Undergraduate Seminars: Toric Varieties Lena Ji February 3, 2016 Contents 1. Algebraic Sets 1 2. The Zariski Topology 3 3. Morphisms of A ne Algebraic Sets 5 4. Dimension 6 References 6 1. Algebraic Sets Let k be an algebraically closed field, for instance the complex numbers C. We define a ne n-space over k to be the set of n-tuples of elements in k. That is, A n = {(a 1,...,a n ) a i 2 k}. We denote by k[x 1,...,x n ] the polynomial ring in n variables with coe cients in k. Polynomials f 2 k[x 1,...,x n ] can be viewed as maps from A n to A by evaluating f at each point, and so we can consider the set of zeroes of a polynomial. More generally, for a collection of polynomials {f i } i2i,wedefinetheirzerosettobe V ({f i } i2i )={(a 1,...,a n ) 2 A n f i (a 1,...,a n )=0 8i 2 I}. Definition 1.1. A subset of A n of the form V ({f i } i2i ) is called an a set. ne algebraic Note that these are referred to as a ne algebraic varieties in Smith, et al. However, we will follow Fulton and adopt the following definition. Definition 1.2. An algebraic set V A n is irreducible if, for any expression V = V 1 [ V 2 where V i are algebraic sets in A n, V 1 = V or V 2 = V. Definition 1.3. An a ne algebraic variety is an irreducible a ne algebraic set. Example 1.4. (1) V (xy) C 2 is an a ne algebraic set, but it is not irreducible. (Figure 1) (2) V (y 2 x 2 x 3 ) C 2 is an a ne algebraic variety. (Figure 2) Here are many more examples of a pictures are over R. Example 1.5. A point (a 1,...,a n ) 2 A n is an a V (x 1 a 1,...,x n a n )={(a 1,...,a n )}. ne algebraic sets. Due to artistic limitations, 1 ne algebraic variety because
2 2 /(,,} f },,)\ Figure 1. V (xy) Figure 2. V (y 2 x 2 x 3 ) Figure 3. V (x 2 + y 2 z 2 ) Figure 4. V (y 2 x(x 2 1)) Example 1.6. A hypersurface in A n is the zero set of a single nonconstant polynomial, for example the quadratic cone V (x 2 + y 2 z 2 ) C 2. A hypersurface in A 2 is called an a ne plane curve. The a ne variety given in Figure 2 is an a ne plane curve, as is the elliptic curve V (y 2 x(x 2 1)). Example 1.7. This very nice heart is a hypersurface given by the solutions of the equation (x y2 + z 2 1) 3 x 2 z y2 z 3 = 0. Example 1.8. The Whitney umbrella is defined by the equation x 2 y 2 z = 0.
3 A ne Algebraic Varieties 3 Example 1.9. The torus with major radius R and minor radius r is defined by the equation (x 2 + y 2 + z 2 + R 2 r 2 ) 2 = R 2 (x 2 + y 2 ). Example The special linear group SL(n, C) ={A 2 M n (C) det(a) =1} is a hypersurface in M n (C) = C n2. This follows from the fact that the determinant is a polynomial in n 2 variables; for example when n = 3, then 0 a b c 1 d e fa = aei + bfg + cdh ceg bdi afh. g h i Example The unit sphere S n 1 C n is an a ne algebraic variety defined by the equation x x 2 n = 1. However, the unit open ball (in the Euclidean topology) and defined as the set {(a 1,...,a n ) 2 C n a a 2 n < 1} is not; if a polynomial vanishes on an open subset of C n in the Euclidean topology, then it is uniformly The Zariski Topology Recall that a collection T of subsets of a space X defines a topology on X if (1) X and ; are in T ; (2) the union of any subcollection of elements of T is contained in T ; (3) the intersection of any finite subcollection of elements of T is in T. We would like to use a ne algebraic sets to define the closed sets of a topology on A n, and so we must check that (1 F ) A n and ; are a ne algebraic sets; (2 F ) the arbitrary intersection of a ne algebraic sets is itself an a ne algebraic set; (3 F ) the finite union of a ne algebraic sets is itself an a ne algebraic set. Let s verify these! If a 2 A is nonzero, then the polynomial equation a = 0 has no solutions, and so V (a) =;. However, the equation 0 = 0 is satisfied by every point in A n, and so V (0) = A n. So condition (1 F ) is satisfied. For (2 F ), let {V } 2A be a collection of a ne algebraic sets, where each V = V ({f i } i 2I ). Then the intersection T 2A V is the common zero set of {f i } i 2I over all 2 A, i.e.! \ [ V = V {f i } i 2I. 2A 2A The twisted cubic curve in Figure 5 illustrates this, as it is given by the intersection of two surfaces: V (x 2 y) \ V (x 3 z)=v (x 2 y, x 3 z). So it remains to show (3 F ). By induction, it is enough to check the union of two a ne algebraic sets.
4 4 /(,,} f },,)\ Figure 5. An intersection Figure 6. A union Proposition 2.1 ([3] Exercise 1.2.1). The union of two a is an a ne algebraic set. ne algebraic sets in A n Proof. Let V ({f i } i2i ) and V ({g j } j2j ) be a ne algebraic sets. We claim that V ({f i } i2i ) [ V ({g j } j2j )=V ({f i g j } (i,j)2i J ). Certainly holds, since if p =(a 1,...,a n ) 2 V ({f i } i2i )[V ({g j } j2j ), then f i (p) = 0 for all i or g j (p) = 0 for all j. In either case, then f i g j (p) = 0 for all i and all j, and so p 2 V ({f i g j } (i,j)2i J ). Now let q 2 V ({f i g j } (i,j)2i J ) and suppose that q 62 V ({f i } i2i ) [ V ({g j } j2j ). Then there exist i and j with f i (q) 6= 0 and g j (q) 6= 0. But this implies f i g j (q) 6= 0, a contradiction, so q must be in V ({f i } i2i )[V ({g j } j2j ) and we have shown. An example of this is the union of the x-axis and yz-plane in Figure 6: V (x) [ V (y, z) =V (xy, xz). Definition 2.2. The topology on A n where the closed sets are of the form V ({f i } i2i ) is called the Zariski topology. The Zariski topology on A n is very di erent from the Euclidean topology. Open subsets in this topology are very big; in fact they are dense and quasi-compact. Additionally, two non-empty open sets will always intersect, and so the Zariski topology is not Hausdor on A n for n>0. Example 2.3. Let k = C. Then the Zariski topology on A 1 is the cofinite topology on C closed sets are ;, C, and finite sets since polynomials in one variable have finitely many roots. Example 2.4 ([3] Exercise 1.2.2). The Zariski topology on A 2 is not the product topology on A 1 A 1. Recall that the product topology on X X is generated by open sets of the form U 1 U 2,whereU 1,U 2 are open subsets of X. SoifA 1 A 1 is equipped with the product topology, where each A 1 has the Zariski topology, open sets are of the form A 2 {finitely many horizontal lines [ vertical lines [ points} The diagonal of A 1 A 1,defined A 1 A 1 = {(a 1,a 2 ) 2 A 1 A 1 a 1 = a 2 }, is not closed in the product topology, where each A 1 is endowed with the Zariski topology, since A 1 is not Hausdor. However, it is the zero set of the polynomial x y 2 A[x, y], so A 1 A 1 = V (x y) A2 is closed in the Zariski topology on A 2.
5 A ne Algebraic Varieties 5 If V A n is an a ne algebraic set, then we can endow V with the subspace topology induced by the Zariski topology on A n. Then closed subsets of V are of the form V \ W,whereW A n is an a ne algebraic set. 3. Morphisms of Affine Algebraic Sets Definition 3.1. Let V A n and W A m be a ne algebraic varieties. A morphism of algebraic varieties is a map F : V! W given by the restriction of a polynomial map A n! A m (meaning that each of the m components is given by a polynomial in k[x 1,...,x n ]). So when we compose F with the inclusion i : W,! A m, the resulting map will be of the form i F =(F 1,...,F m ) where each F i is the restriction to V of a (non-unique) polynomial in k[x 1,...,x n ]. Definition 3.2. A morphism F : V! W is an isomorphism if it has an inverse morphism. In this case we say that V and W are isomorphic. Example 3.3. Let C be the plane parabola given by the equation y x 2 = 0. Then C is isomorphic to A 1 via the maps ' C, where ' : A 2! A 1 A 1! C (x, y) 7! x t 7! (t, t 2 ). Example 3.4 ([3] Exercise 1.3.2). The twisted cubic V = V (x 2 y, x 3 z)in Figure 5 is isomorphic to A 1.Since V = {(t, t 2,t 3 ) 2 A 3 t 2 A}, we can define a morphism A 1! V by t 7! (t, t 2,t 3 ). The restriction to V of the projection A 3! A 1 onto the first factor, defined (x, y, z) 7! x, gives an inverse morphism. Proposition 3.5 ([3] Exercise 1.3.1). If F : V! W is a morphism of a ne algebraic sets, then F is continuous in the Zariski topology. Proof. Any closed subset of W is of the form V ({f i } i2i )\W where f i 2 k[x 1,...,x m ]. F 1 (V ({f i } i2i ) \ W )=F 1 (V ({f i } i2i )) \ F 1 (W )=V ({f i F }i2i ) \ V is a closed subset of V,where F : A n to F,soF is continuous.! A m is any polynomial map that restricts
6 6 /(,,} f },,)\ 4. Dimension Definition 4.1. The dimension of an a ne algebraic set V is the length of the longest chain of distinct nonempty a ne closed subvarieties of V sup{d V d ) V d 1 ) ) V 0 }. Hence the dimension of an a ne algebraic set is equal to the maximum of the dimensions of its irreducible components (maximal irreducible subsets). Example 4.2. The quadratic cone V (x 2 + y 2 z 2 ) has dimension 2. Example 4.3. The a ne algebraic set V (xy, xz) has dimension 2 = max{1, 2}. Definition 4.4. An a ne algebraic set is equidimensional if all of its irreducible components have the same dimension. So in the earlier examples, V (xy, xz) is not equdimensional but V (x 2 + y 2 z 2 ) and V (xy) (Figure 1) are. Definition 4.5. The codimension of an a codim V = n dim V. ne algebraic sets of codi- Example 4.6. Hypersurfaces in A n are precisely the a mension 1. References ne algebraic set V A n is defined [1] William Fulton. Algebraic Curves, [2] Herwig Hauser. Algebraic Surfaces Gallery. bildergalerie/gallery.html. [3] Karen Smith, Lauri Kahanpää, Pekka Kekäläinen, and William Traves. An Invitation to Algebraic Geometry. Springer,2004.
Summer Algebraic Geometry Seminar
Summer Algebraic Geometry Seminar Lectures by Bart Snapp About This Document These lectures are based on Chapters 1 and 2 of An Invitation to Algebraic Geometry by Karen Smith et al. 1 Affine Varieties
More informationMath 203A - Solution Set 1
Math 203A - Solution Set 1 Problem 1. Show that the Zariski topology on A 2 is not the product of the Zariski topologies on A 1 A 1. Answer: Clearly, the diagonal Z = {(x, y) : x y = 0} A 2 is closed in
More informationCHEVALLEY 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
More informationMath 418 Algebraic Geometry Notes
Math 418 Algebraic Geometry Notes 1 Affine Schemes Let R be a commutative ring with 1. Definition 1.1. The prime spectrum of R, denoted Spec(R), is the set of prime ideals of the ring R. Spec(R) = {P R
More informationMath 203A - Solution Set 1
Math 203A - Solution Set 1 Problem 1. Show that the Zariski topology on A 2 is not the product of the Zariski topologies on A 1 A 1. Answer: Clearly, the diagonal Z = {(x, y) : x y = 0} A 2 is closed in
More informationMath 145. Codimension
Math 145. Codimension 1. Main result and some interesting examples In class we have seen that the dimension theory of an affine variety (irreducible!) is linked to the structure of the function field in
More informationAlgebraic Geometry of Matrices II
Algebraic Geometry of Matrices II Lek-Heng Lim University of Chicago July 3, 2013 today Zariski topology irreducibility maps between varieties answer our last question from yesterday again, relate to linear
More informationMath 203A - Solution Set 1
Math 203A - Solution Set 1 Problem 1. Show that the Zariski topology on A 2 is not the product of the Zariski topologies on A 1 A 1. Answer: Clearly, the diagonal Z = {(x, y) : x y = 0} A 2 is closed in
More information10. 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
More informationResolution 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.
More informationALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!
ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.
More informationMath 203A, Solution Set 6.
Math 203A, Solution Set 6. Problem 1. (Finite maps.) Let f 0,..., f m be homogeneous polynomials of degree d > 0 without common zeros on X P n. Show that gives a finite morphism onto its image. f : X P
More informationCOMPLEX 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
More informationMATH 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
More informationInstitutionen för matematik, KTH.
Institutionen för matematik, KTH. Contents 7 Affine Varieties 1 7.1 The polynomial ring....................... 1 7.2 Hypersurfaces........................... 1 7.3 Ideals...............................
More information(dim Z j dim Z j 1 ) 1 j i
Math 210B. Codimension 1. Main result and some interesting examples Let k be a field, and A a domain finitely generated k-algebra. In class we have seen that the dimension theory of A is linked to the
More informationAlgebraic 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
More informationπ X : X Y X and π Y : X Y Y
Math 6130 Notes. Fall 2002. 6. Hausdorffness and Compactness. We would like to be able to say that all quasi-projective varieties are Hausdorff and that projective varieties are the only compact varieties.
More informationExercise Sheet 7 - Solutions
Algebraic Geometry D-MATH, FS 2016 Prof. Pandharipande Exercise Sheet 7 - Solutions 1. Prove that the Zariski tangent space at the point [S] Gr(r, V ) is canonically isomorphic to S V/S (or equivalently
More information3. 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
More information9. Birational Maps and Blowing Up
72 Andreas Gathmann 9. Birational Maps and Blowing Up In the course of this class we have already seen many examples of varieties that are almost the same in the sense that they contain isomorphic dense
More informationwhere 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
More informationINTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 14
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 14 RAVI VAKIL Contents 1. Dimension 1 1.1. Last time 1 1.2. An algebraic definition of dimension. 3 1.3. Other facts that are not hard to prove 4 2. Non-singularity:
More informationMATH32062 Notes. 1 Affine algebraic varieties. 1.1 Definition of affine algebraic varieties
MATH32062 Notes 1 Affine algebraic varieties 1.1 Definition of affine algebraic varieties We want to define an algebraic variety as the solution set of a collection of polynomial equations, or equivalently,
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 4: MORE ABOUT VARIETIES AND REGULAR FUNCTIONS.
ALGERAIC GEOMETRY COURSE NOTES, LECTURE 4: MORE AOUT VARIETIES AND REGULAR FUNCTIONS. ANDREW SALCH. More about some claims from the last lecture. Perhaps you have noticed by now that the Zariski topology
More informationPROBLEMS, MATH 214A. Affine and quasi-affine varieties
PROBLEMS, MATH 214A k is an algebraically closed field Basic notions Affine and quasi-affine varieties 1. Let X A 2 be defined by x 2 + y 2 = 1 and x = 1. Find the ideal I(X). 2. Prove that the subset
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 Throughout this lecture k denotes an algebraically closed field. 17.1 Tangent spaces and hypersurfaces For any polynomial f k[x
More informationExploring the Exotic Setting for Algebraic Geometry
Exploring the Exotic Setting for Algebraic Geometry Victor I. Piercey University of Arizona Integration Workshop Project August 6-10, 2010 1 Introduction In this project, we will describe the basic topology
More informationCHAPTER 1. AFFINE ALGEBRAIC VARIETIES
CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the
More informationD-MATH Algebraic Geometry FS 2018 Prof. Emmanuel Kowalski. Solutions Sheet 1. Classical Varieties
D-MATH Algebraic Geometry FS 2018 Prof. Emmanuel Kowalski Solutions Sheet 1 Classical Varieties Let K be an algebraically closed field. All algebraic sets below are defined over K, unless specified otherwise.
More informationPure Math 764, Winter 2014
Compact course notes Pure Math 764, Winter 2014 Introduction to Algebraic Geometry Lecturer: R. Moraru transcribed by: J. Lazovskis University of Waterloo April 20, 2014 Contents 1 Basic geometric objects
More informationCurtis Heberle MTH 189 Final Paper 12/14/2010. Algebraic Groups
Algebraic Groups Curtis Heberle MTH 189 Final Paper 12/14/2010 The primary objects of study in algebraic geometry are varieties. Having become acquainted with these objects, it is interesting to consider
More informationMAT4210 Algebraic geometry I: Notes 2
MAT4210 Algebraic geometry I: Notes 2 The Zariski topology and irreducible sets 26th January 2018 Hot themes in notes 2: The Zariski topology on closed algebraic subsets irreducible topological spaces
More informationYuriy Drozd. Intriduction to Algebraic Geometry. Kaiserslautern 1998/99
Yuriy Drozd Intriduction to Algebraic Geometry Kaiserslautern 1998/99 CHAPTER 1 Affine Varieties 1.1. Ideals and varieties. Hilbert s Basis Theorem Let K be an algebraically closed field. We denote by
More informationNONSINGULAR 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
More information4. 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
More informationVector 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,
More informationAlgebraic Geometry. Andreas Gathmann. Notes for a class. taught at the University of Kaiserslautern 2002/2003
Algebraic Geometry Andreas Gathmann Notes for a class taught at the University of Kaiserslautern 2002/2003 CONTENTS 0. Introduction 1 0.1. What is algebraic geometry? 1 0.2. Exercises 6 1. Affine varieties
More information14. 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
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More information10. Noether Normalization and Hilbert s Nullstellensatz
10. Noether Normalization and Hilbert s Nullstellensatz 91 10. Noether Normalization and Hilbert s Nullstellensatz In the last chapter we have gained much understanding for integral and finite ring extensions.
More informationAlgebraic Geometry. Andreas Gathmann. Class Notes TU Kaiserslautern 2014
Algebraic Geometry Andreas Gathmann Class Notes TU Kaiserslautern 2014 Contents 0. Introduction......................... 3 1. Affine Varieties........................ 9 2. The Zariski Topology......................
More informationELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS
ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need
More informationAlgebraic 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
More informationIntroduction to Algebraic Geometry. Jilong Tong
Introduction to Algebraic Geometry Jilong Tong December 6, 2012 2 Contents 1 Algebraic sets and morphisms 11 1.1 Affine algebraic sets.................................. 11 1.1.1 Some definitions................................
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then
More informationdiv(f ) = D and deg(d) = deg(f ) = d i deg(f i ) (compare this with the definitions for smooth curves). Let:
Algebraic Curves/Fall 015 Aaron Bertram 4. Projective Plane Curves are hypersurfaces in the plane CP. When nonsingular, they are Riemann surfaces, but we will also consider plane curves with singularities.
More information8. Prime Factorization and Primary Decompositions
70 Andreas Gathmann 8. Prime Factorization and Primary Decompositions 13 When it comes to actual computations, Euclidean domains (or more generally principal ideal domains) are probably the nicest rings
More informationLecture 1. Toric Varieties: Basics
Lecture 1. Toric Varieties: Basics Taras Panov Lomonosov Moscow State University Summer School Current Developments in Geometry Novosibirsk, 27 August1 September 2018 Taras Panov (Moscow University) Lecture
More informationReid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed.
Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed. Answer: Note that the first generator factors as (y
More informationThe Grothendieck Ring of Varieties
The Grothendieck Ring of Varieties Ziwen Zhu University of Utah October 25, 2016 These are supposed to be the notes for a talk of the student seminar in algebraic geometry. In the talk, We will first define
More informationV (f) :={[x] 2 P n ( ) f(x) =0}. If (x) ( x) thenf( x) =
20 KIYOSHI IGUSA BRANDEIS UNIVERSITY 2. Projective varieties For any field F, the standard definition of projective space P n (F ) is that it is the set of one dimensional F -vector subspaces of F n+.
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1)
Tuesday 10 February 2004 (Day 1) 1a. Prove the following theorem of Banach and Saks: Theorem. Given in L 2 a sequence {f n } which weakly converges to 0, we can select a subsequence {f nk } such that the
More informationne 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
More informationSecant varieties. Marin Petkovic. November 23, 2015
Secant varieties Marin Petkovic November 23, 2015 Abstract The goal of this talk is to introduce secant varieies and show connections of secant varieties of Veronese variety to the Waring problem. 1 Secant
More informationThis is a closed subset of X Y, by Proposition 6.5(b), since it is equal to the inverse image of the diagonal under the regular map:
Math 6130 Notes. Fall 2002. 7. Basic Maps. Recall from 3 that a regular map of affine varieties is the same as a homomorphism of coordinate rings (going the other way). Here, we look at how algebraic properties
More informationA course in. Algebraic Geometry. Taught by Prof. Xinwen Zhu. Fall 2011
A course in Algebraic Geometry Taught by Prof. Xinwen Zhu Fall 2011 1 Contents 1. September 1 3 2. September 6 6 3. September 8 11 4. September 20 16 5. September 22 21 6. September 27 25 7. September
More informationMath 6140 Notes. Spring Codimension One Phenomena. Definition: Examples: Properties:
Math 6140 Notes. Spring 2003. 11. Codimension One Phenomena. A property of the points of a variety X holds in codimension one if the locus of points for which the property fails to hold is contained in
More informationExercise Sheet 3 - Solutions
Algebraic Geometry D-MATH, FS 2016 Prof. Pandharipande Exercise Sheet 3 - Solutions 1. Prove the following basic facts about algebraic maps. a) For f : X Y and g : Y Z algebraic morphisms of quasi-projective
More informationLECTURE Affine Space & the Zariski Topology. It is easy to check that Z(S)=Z((S)) with (S) denoting the ideal generated by elements of S.
LECTURE 10 1. Affine Space & the Zariski Topology Definition 1.1. Let k a field. Take S a set of polynomials in k[t 1,..., T n ]. Then Z(S) ={x k n f(x) =0, f S}. It is easy to check that Z(S)=Z((S)) with
More informationProjective Varieties. Chapter Projective Space and Algebraic Sets
Chapter 1 Projective Varieties 1.1 Projective Space and Algebraic Sets 1.1.1 Definition. Consider A n+1 = A n+1 (k). The set of all lines in A n+1 passing through the origin 0 = (0,..., 0) is called the
More informationLECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are
More informationMath 203A - Solution Set 3
Math 03A - Solution Set 3 Problem 1 Which of the following algebraic sets are isomorphic: (i) A 1 (ii) Z(xy) A (iii) Z(x + y ) A (iv) Z(x y 5 ) A (v) Z(y x, z x 3 ) A Answer: We claim that (i) and (v)
More informationCartan s Criteria. Math 649, Dan Barbasch. February 26
Cartan s Criteria Math 649, 2013 Dan Barbasch February 26 Cartan s Criteria REFERENCES: Humphreys, I.2 and I.3. Definition The Cartan-Killing form of a Lie algebra is the bilinear form B(x, y) := Tr(ad
More informationINTERSECTION THEORY CLASS 2
INTERSECTION THEORY CLASS 2 RAVI VAKIL CONTENTS 1. Last day 1 2. Zeros and poles 2 3. The Chow group 4 4. Proper pushforwards 4 The webpage http://math.stanford.edu/ vakil/245/ is up, and has last day
More informationAlgebraic Geometry (Math 6130)
Algebraic Geometry (Math 6130) Utah/Fall 2016. 2. Projective Varieties. Classically, projective space was obtained by adding points at infinity to n. Here we start with projective space and remove a hyperplane,
More informationHere 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
More informationAN 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,
More informationAlgebraic Varieties. Brian Osserman
Algebraic Varieties Brian Osserman Preface This book is largely intended as a substitute for Chapter I (and an invitation to Chapter IV) of Hartshorne [Har77], to be taught as an introduction to varieties
More informationThe Geometry-Algebra Dictionary
Chapter 1 The Geometry-Algebra Dictionary This chapter is an introduction to affine algebraic geometry. Working over a field k, we will write A n (k) for the affine n-space over k and k[x 1,..., x n ]
More informationMAS 6396 Algebraic Curves Spring Semester 2016 Notes based on Algebraic Curves by Fulton. Timothy J. Ford April 4, 2016
MAS 6396 Algebraic Curves Spring Semester 2016 Notes based on Algebraic Curves by Fulton Timothy J. Ford April 4, 2016 FLORIDA ATLANTIC UNIVERSITY, BOCA RATON, FLORIDA 33431 E-mail address: ford@fau.edu
More informationEIGENVALUES AND EIGENVECTORS 3
EIGENVALUES AND EIGENVECTORS 3 1. Motivation 1.1. Diagonal matrices. Perhaps the simplest type of linear transformations are those whose matrix is diagonal (in some basis). Consider for example the matrices
More information(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
More informationB 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.
Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2
More informationRings and groups. Ya. Sysak
Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...
More informationALGEBRAIC GEOMETRY (NMAG401) Contents. 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30
ALGEBRAIC GEOMETRY (NMAG401) JAN ŠŤOVÍČEK Contents 1. Affine varieties 1 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30 1. Affine varieties The basic objects
More information4.4 Noetherian Rings
4.4 Noetherian Rings Recall that a ring A is Noetherian if it satisfies the following three equivalent conditions: (1) Every nonempty set of ideals of A has a maximal element (the maximal condition); (2)
More informationMath 203A - Solution Set 4
Math 203A - Solution Set 4 Problem 1. Let X and Y be prevarieties with affine open covers {U i } and {V j }, respectively. (i) Construct the product prevariety X Y by glueing the affine varieties U i V
More informationAlgebraic varieties and schemes over any scheme. Non singular varieties
Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two
More informationADVANCED TOPICS IN ALGEBRAIC GEOMETRY
ADVANCED TOPICS IN ALGEBRAIC GEOMETRY DAVID WHITE Outline of talk: My goal is to introduce a few more advanced topics in algebraic geometry but not to go into too much detail. This will be a survey of
More informationAlgebraic Varieties. Chapter Algebraic Varieties
Chapter 12 Algebraic Varieties 12.1 Algebraic Varieties Let K be a field, n 1 a natural number, and let f 1,..., f m K[X 1,..., X n ] be polynomials with coefficients in K. Then V = {(a 1,..., a n ) :
More information2. Intersection Multiplicities
2. Intersection Multiplicities 11 2. Intersection Multiplicities Let us start our study of curves by introducing the concept of intersection multiplicity, which will be central throughout these notes.
More information12. 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
More informationGLUING SCHEMES AND A SCHEME WITHOUT CLOSED POINTS
GLUING SCHEMES AND A SCHEME WITHOUT CLOSED POINTS KARL SCHWEDE Abstract. We first construct and give basic properties of the fibered coproduct in the category of ringed spaces. We then look at some special
More informationAlgebraic Geometry. Question: What regular polygons can be inscribed in an ellipse?
Algebraic Geometry Question: What regular polygons can be inscribed in an ellipse? 1. Varieties, Ideals, Nullstellensatz Let K be a field. We shall work over K, meaning, our coefficients of polynomials
More informationAPPENDIX 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
More information11. Dimension. 96 Andreas Gathmann
96 Andreas Gathmann 11. Dimension We have already met several situations in this course in which it seemed to be desirable to have a notion of dimension (of a variety, or more generally of a ring): for
More informationPart III. 10 Topological Space Basics. Topological Spaces
Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.
More informationTHE ALGEBRAIC GEOMETRY DICTIONARY FOR BEGINNERS. Contents
THE ALGEBRAIC GEOMETRY DICTIONARY FOR BEGINNERS ALICE MARK Abstract. This paper is a simple summary of the first most basic definitions in Algebraic Geometry as they are presented in Dummit and Foote ([1]),
More information9. Integral Ring Extensions
80 Andreas Gathmann 9. Integral ing Extensions In this chapter we want to discuss a concept in commutative algebra that has its original motivation in algebra, but turns out to have surprisingly many applications
More informationSmooth 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,
More information2. Prime and Maximal Ideals
18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the so-called prime and maximal ideals. Let
More information3. 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)
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013 As usual, all the rings we consider are commutative rings with an identity element. 18.1 Regular local rings Consider a local
More informationAN INTRODUCTION TO TORIC SURFACES
AN INTRODUCTION TO TORIC SURFACES JESSICA SIDMAN 1. An introduction to affine varieties To motivate what is to come we revisit a familiar example from high school algebra from a point of view that allows
More informationChapter 8. P-adic numbers. 8.1 Absolute values
Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.
More information1 Directional Derivatives and Differentiability
Wednesday, January 18, 2012 1 Directional Derivatives and Differentiability Let E R N, let f : E R and let x 0 E. Given a direction v R N, let L be the line through x 0 in the direction v, that is, L :=
More information16 Dimension and transcendence degree
16 Dimension and transcendence degree 16.A Dimension: desired properties Our goal is to define the dimension of an algebraic variety over k. It should be a welldefinedfunction from the setof isomorphism
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets
FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be
More information