# k k would be reducible. But the zero locus of f in A n+1

Size: px
Start display at page:

Download "k k would be reducible. But the zero locus of f in A n+1"

Transcription

1 Math 145. Bezout s Theorem Let be an algebraically closed field. The purpose of this handout is to prove Bezout s Theorem and some related facts of general interest in projective geometry that arise along the way. 1. Setup and projective transformations Let f, g [X, Y, Z] be homogenous of respective degrees d, e 1, without any common irreducible factor. The zero loci Z(f), Z(g) have as their respective sets of irreducible components the zero loci of the (homogenous!) irreducible factors {f i } of f and {g j } of g due to: Proposition 1.1. If f [x 0,..., x n ] is homogeneous of degree d > 0 then all factors of f in [x 0,..., x n ] are homogeneous, and if f is irreducible then the closed set Z(f) P n is irreducible of dimension n 1. If g is another homogeneous irreducible polynomial in [x 0,..., x n ] then Z(f) = Z(g) if and only if f = cg for c. Proof. If f = f 1 f 2 with non-constant f j s then the degrees d j > 0 of the f j s must add up to d and the top-degree parts of the f j s have product then must be the top degree part of f. But f is its own top-degree part (as f is homogeneous). By similar reasoning, the least-degree parts of the f j s must have product equal to f and in particular degrees that add up to d. These least degrees are respectively d 1 and d 2, yet their sum is d = d 1 + d 2, so the least-degrees coincide with the top degrees for the f j s; i.e., each f j is homogeneous. Next assume that f is irreducible. To show that Z(f) is irreducible, consider the surjective morphism q : A n+1 {0} P n. The preimage is the zero locus of f on A n+1 with the origin removed. If Z(f) is covered by two proper closed subsets in P n, then the zero locus of f in An+1 {0} would be covered by their proper closed preimages. That is, the zero locus of f in the open subset Ω = A n+1 {0} A n+1 would be reducible. But the zero locus of f in A n+1 is irreducible since f is irreducible, and it contains the origin as a proper closed subset (since the origin is of codimension n + 1 2, so it cannot be an irreducible hypersurface), so its overlap with A n+1 {0} is a non-empty open subset of an irreducible space. Such an open subset is always irreducible. The zero locus V of f in A n+1 {0} is irreducible of dimension n, and it maps onto Z(f) with all fibers of dimension 1. More specifically, each q 1 (U i ) U i is a surjective map between affine varieties with all fibers irreducible of dimension 1, so liewise whenever Z(f) U i is non-empty we see that V q 1 (U i ) Z(f) U i is a surjective map between affine varieties with all fibers of dimension 1. Thus, by the earlier handout on fiber dimension in the affine case, it follows that the target Z(f) U i has dimension dim(v q 1 (U i )) 1 = n 1. Finally, for irreducible homogenous g [x 0,..., x n ] such that Z(g) = Z(f) we see to prove that g = cf for some c. Consider the preimages in A n+1 {0}. These preimages are the complements of the origin in the irreducible (!) zero loci of f and g in A n+1, so the closures of these preimages in A n+1 are those zero loci. That is, f and g have the same zero loci in A n+1, whence each is a multiple of the other in [x 0,..., x n ] by their irreducibility and the Nullstellensatz. This forces the multiplier relating them to be a unit in [x 0,..., x n ], which is to say an element of c. The hypotheses on f and g ensure that the irreducible curves Z(f i ), Z(g j ) are distinct, whence have intersection of smaller dimension, whence finite! For non-constant polynomials h 1, h 2 [u, v] and points P not on a common irreducible component of Z(h 1 ) and Z(h 2 ) (i.e., an isolated point of Z(h 1, h 2 ) or a point not in this common zero locus at all) our definition of the intersection number I(P ; h 1, h 2 ) = dim (O A 2,P /(h 1, h 2 )) < (which vanishes if one of h 1 or h 2 is nonzero at P ) is given in terms of the local rings (with h 1, h 2 [u, v] = O(A 2 ) having no common irreducible factor through P ). Thus, we can define intersection numbers I(P ; f, g) for homogenous f, g and any point P by simply dehomogenizing f and g with respect to any of the homogenous coordinates not vanishing at P and then computing the (finite!) intersection number in 1

2 2 the corresponding A 2 via our affine theory. Note that if P has more than one non-vanishing coordinate then it does not matter which coordinate we use for dehomogenization: for example, if P U 0 U 2 with P = [1, a, b] = [1/b, a/b, 1] for b 0 then we have an isomorphism O A 2,(a,b) = O U0,P = O P 2,P = O U2,P = O A 2,(1/b,a/b) which carries the X-dehomogenizations of h 1 and h 2 over to the Z-dehomogenizations of h 1 and h 2 respectively, so passing to the respective quotients by these identifies (as local -algebras with finite dimension as -vector spaces) the two local ring quotients. It is important to chec that that this well-posed definition is unaffected by arbitrary projective linear change of coordinates, (not just the most trivial type which interchange the coordinates). That is, for T PGL 3 () and T GL 3 () representing T, we claim that I(P ; f, g) = I(T 1 (P ), f T, g T ). We give a computation-free proof below (as any other method appears tedious). Remar 1.2. Intersection numbers can be defined entirely intrinsically i.e., without maing reference to coordinates or A 2 s within the framewor of schemes. The fact that we built our theory out of reduced -algebras forces us to use the above ad hoc approach, roughly because we cannot define the geometric object Z(f) in such a way that it nows the multiplicites of the irreducible factors of f (if such multiplicities occur which are > 1). Lemma 1.3. The above definition of I(P ; f, g) is unaffected by projective linear change of coordinates. Proof. From the affine case, it follows that our definition is additive with respect to taing products in f and g. That is, if f = c f ri i and g = c g sj j is the decomposition into irreducible (homogenous!) factors (with c, c ), then I(P ; f, g) = i,j r i s j I(P ; f i, g j ) (where the only non-zero terms on the right side are the ones for which f i (P ) = g j (P ) = 0). Since the formation of the irreducible factorization behaves well under linear homogenous change of coordinates, we are therefore reduced to studying the case in which f and g are irreducible. The point of this step is that the closed subvarieties Z(f) and Z(g) remember the irreducible f and g, by Proposition 1.1. For irreducible homogenous f and g which aren t scalar multiples of each other (i.e., Z(f) Z(g)), we now give a description of I(P ; f, g) which depends only on the closed subvarieties Z(f), Z(g) P = P 2 and not on any choice of coordinate system. In general, let Σ be any irreducible surface (i.e., 2-dimensional variety), and C, C two distinct irreducible curves on Σ, so C C is a finite set. Choose P C C, and consider the surjections O Σ,P O C,P, O C,P (why are these surjective?). Let I P, I P denote the respective ernels, and define J P = I P + I P, which is morally the ideal defining C C at P (except that this ideal might not be radical, which corresponds to the possibility of an intersection number being > 1). For example, if Σ = A 2 and C = Z(f), C = Z(f ) for irreducible f, f and P = (0, 0), then I P = fo A 2,0, I P = f O A 2,0, and J P = (f, f )O A 2,0. This example naturally leads us to consider the quotient -algebra O Σ,P /J P. In the case of distinct irreducible curves in A 2 passing through a point, this quotient -algebra is exactly the one whose -dimension is the finite integer which we defined to be I(P ; f, g)! We conclude that with Σ = P 2, C = Z(f), C = Z(g) where f and g are homogenous and irreducible (whence have irreducible dehomogenizations with respect to any of the coordinate variables which is non-vanishing at a point P C C ), the intrinsic -algebra O Σ,P /J P is of finite -dimension equal to the coordinate-dependently defined I(P ; f, g). The intrinsic nature of this local ring quotient (its definition involves no specification of coordinate, homogenous or otherwise!) thereby implies that this integer is in fact unaffected by arbitrary projective linear change of coordinates on P 2.

3 3 Here is Bezout s theorem: 2. Main result Theorem 2.1. For f, g [X, Y, Z] homogenous of respective positive degrees d, e and without any common irreducible factor, I(P ; f, g) = de. P P 2 We emphasize that there are only finitely many non-zero terms in the sum, corresponding to the finite set Z(f) Z(g). Proof. By Lemma 1.3, we may mae any linear homogenous change of coordinates. Also, using additivity of both intersection numbers and degrees with respect to products in f and g (as in the proof of Lemma 1.3), we immediately reduce to the case where f and g are irreducible. Since Z(f) Z(g) is a finite set, we may find a line P 1 P2 disjoint from this finite set (how?), and may then apply a projective linear change of coordinates so that this line is the line at infinity, Z = 0. That is, we can assume the intersection points all lie in the ordinary affine plane {Z 0} = U 2. Let S = [X, Y, Z], on which we have a natural direct sum decomposition S = r 0 S r with S r the space of homogeneous polynomials of degree r. An ideal I S is called homogeneous if I = I r with I r = I S r ; i.e., I contains all homogeneous parts of all of its elements. The quotient S/I = r 0 (S r /I r ) by a homogeneous ideal inherits a natural notion of homogeneous part (namely, the spaces S r /I r, or equivalently the image of S r in S/I for each r 0). It is clear that a sum of homogeneous ideals is homogeneous (chec!), and that if h S is homogeneous then the principal ideal (h) = hs is homogeneous (why?). Thus, (f), (g), and (f, g) = (f) + (g) are homogeneous ideals, so each of the quotients S/f, S/g, S/(f, g) has a natural notion of homogeneous parts. The homogeneous parts of each are quotients of the S r s and so have finite -dimension as vector spaces. Since f and g are irreducibles in the UFD S and are not unit multiples of each other, the sequence of -vector spaces 0 S/g f S/g S/(f, g) 0 is exact. Note that although the second map respects the notion of degree of homogeneous part induced from S, the first map shifts such degrees up by the degree d of the homogenous f. If we let ( ) r denote the degree-rth homogeneous part of the ring then we have an exact sequence 0 (S/g) r d f (S/g)r (S/(f, g)) r 0 (where the pieces in negative degrees are understood to be 0). Thus, dim (S/(f, g)) r = dim (S/g) r dim (S/g) r d. Using the exact sequence 0 S g S S/g 0 and similar considerations with grading (where now the first map shifts up degrees by the degree e of g), we deduce dim (S/g) n = dim S n dim S n e. But for m 0 we have dim S m = m(m 1)/2. Note that this applies to S n e only for n e. Inserting this into the above formulae, we deduce dim (S/(f, g)) r = de for r sufficiently large (more precisely, for r e + d). The idea is now to show that for r sufficiently large, (S/(f, g)) r has dimension equal to the sum of the intersection numbers. Roughly speaing, we want to view S/(f, g) as the coordinate ring of the necessarily affine closed subscheme Z(f) Z(g) P 2 {Z = 0} A 2,

4 4 but to mae this precise we will need to be attentive to the fact that elements of S are not really functions on P 2 and moreover the correct way to define this intersection Z(f) Z(g) as a geometric object requires allowing nilpotents in the structure sheaf (this corresponds to points P with I(P ; f, g) > 1). Nonetheless, once such a scheme-theoretic intersection is defined (which we will do here without requiring the concept of a scheme), the -dimension of its coordinate ring ought to be that sum of intersection numbers. Anyway, this is the geometric motivation (which comes from thining about schemes and modern intersection theory). Now consider the dehomogenizations f Z, g Z [x, y]. We claim that the sum I(P ; f, g) = I(P ; f Z, g Z ) P P 2 P A 2 is equal to dim [x, y]/(f Z, g Z ). Since f Z and g Z have no common irreducible factor in [x, y], the method of proof in 2 of the handout on intersection numbers shows that the natural map of -algebras [x, y]/(f Z, g Z ) [x, y] mp /(f Z, g Z ). f Z (P )=g Z (P )=0 For a common zero P of f Z and g Z, the intersection number I(P ; f Z, g Z ) is exactly the -dimension of the P th factor in this direct product, so the sum of these -dimensions over all such P is the common -dimension of both sides of this isomorphism of -algebras, which in turn is visibly dim [x, y]/(f Z, g Z ). We have proven that the two sides of Bezout s theorem are equal to the -dimensions of [x, y]/(f Z, g Z ) and (S/(f, g)) r (for large r) respectively. Now in general for any ideal I in [x, y], define I h S to be the radical homogenous ideal corresponding to the Zarisi closure of Z(I) A 2 = P2 {Z = 0} in P2. This closure meets the open A 2 P2 in Z(I), so I is exactly the set of f [x, y] whose Z-homogenization lies in I h. What is more important is that if F S and ZF I h then F I h (as is easy to chec from the definition of I h in terms of Z-homogenization and Z-dehomogenization). The ring A = [x, y]/i has finite -dimension, and I claim that any finite -algebra B admits a filtration by ideals I j with 0 = I 0 I 1 I N = B with dim I j /I j 1 = 1 for all 0 < j N. Indeed, since B breas up into a finite product of finite local -algebras, we only need to consider local B. If B = there s nothing to say. Otherwise, since B/m B = (as is algebraically closed!!), we must have m B 0, so there is some integer n 1 such that m n B 0, = 0. Choose any non-zero b m n B, so the non-zero principal ideal (b) is illed by m B and hence is of dimension 1 as a -vector space, Applying induction on -dimension to B/b, we get the desired chain of ideals. Now fix such a rising chain of ideals {I j } in A = [x, y]/i, each of -codimension 1 in the next, so dim A = dim I j /I j 1. The rising chain of ideals {I j } in A gives rise to a rising chain of homogenous m n+1 B ideals Ij h in S/(f, g) (here I am implicitly identified ideals in a quotient ring with certain ideals in the original ring), whence (S/(f, g)) r has a filtration by the rising chain of subspaces (Ij h/ih j 1 ) r. We will now prove that if r is sufficiently large, then dim (I h j /I h j 1) r = 1 for all j. Taing the sum over j (with large r) will then complete the proof. More generally, if J K is an inclusion of ideals in [x, y] with dim K/J = 1, we claim that dim (K h /J h ) r = 1 for large r (and hence the same conclusion applies for sufficiently large r for any finite rising chain of such ideals, so we d be done). Let n N = K h /J h be a homogenous element represented by a homogenous element F K h of some degree r. Since the multiplication map by Z on N = K h /J h is injective (why?), we see that F/Z r = F Z J if and only if F / J h, which is equivalent to n 0. Since every element in K has the form F Z for some homogenous F K h and K J, we conclude that N 0. Now choose homogenous n N non-zero with minimal degree d (note that by its definition, N has vanishing graded terms in negative degrees) and choose a homogenous degree d representative F for n as above. For any

5 d d and any homogenous n N d represented by a homogenous F, since K/J has -dimension 1 with basis given by F Z, we conclude that F Z cf Z mod J for some c, whence Z m (Z d F cz d F ) J h for a large integer m. But this implies F cz d d F mod J h (as multiplication by Z is injective on K h /J h ), so n = cz d d n in N d. Thus, for d d the non-zero (!) element Z d d n N d is a -basis, so indeed N r = (K h /J h ) r has -dimension equal to 1 for all large r (i.e., r d). This is what we needed to prove. 5

### Math 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

### Math 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

### 12. Hilbert Polynomials and Bézout s Theorem

12. Hilbert Polynomials and Bézout s Theorem 95 12. Hilbert Polynomials and Bézout s Theorem After our study of smooth cubic surfaces in the last chapter, let us now come back to the general theory of

### (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

### 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

### 2. 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

### 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

### Math 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

### 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,

### 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

### CHEVALLEY S THEOREM AND COMPLETE VARIETIES

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

### Institutionen 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...............................

### ALGEBRAIC 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

### DIVISORS ON NONSINGULAR CURVES

DIVISORS ON NONSINGULAR CURVES BRIAN OSSERMAN We now begin a closer study of the behavior of projective nonsingular curves, and morphisms between them, as well as to projective space. To this end, we introduce

### ALGEBRAIC 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.

### Algebraic 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,

### 10. 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.

### 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.

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

### Projective 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

### CHAPTER 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

### Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime

### 8. 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

### 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

### Algebraic 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 ) :

### ELEMENTARY 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

### MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

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

### ALGEBRAIC GEOMETRY CAUCHER BIRKAR

ALGEBRAIC GEOMETRY CAUCHER BIRKAR Contents 1. Introduction 1 2. Affine varieties 3 Exercises 10 3. Quasi-projective varieties. 12 Exercises 20 4. Dimension 21 5. Exercises 24 References 25 1. Introduction

### 9. 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

### 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

### 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

### Resolution of Singularities in Algebraic Varieties

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

### 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

### π 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.

### 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

### V (v i + W i ) (v i + W i ) is path-connected and hence is connected.

Math 396. Connectedness of hyperplane complements Note that the complement of a point in R is disconnected and the complement of a (translated) line in R 2 is disconnected. Quite generally, we claim that

### TROPICAL SCHEME THEORY

TROPICAL SCHEME THEORY 5. Commutative algebra over idempotent semirings II Quotients of semirings When we work with rings, a quotient object is specified by an ideal. When dealing with semirings (and lattices),

### Introduction 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

### 10. Smooth Varieties. 82 Andreas Gathmann

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

### Algebraic Geometry. Instructor: Stephen Diaz & Typist: Caleb McWhorter. Spring 2015

Algebraic Geometry Instructor: Stephen Diaz & Typist: Caleb McWhorter Spring 2015 Contents 1 Varieties 2 1.1 Affine Varieties....................................... 2 1.50 Projective Varieties.....................................

### 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

### Math 249B. Geometric Bruhat decomposition

Math 249B. Geometric Bruhat decomposition 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ Φ, and let B be the unique

### (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

### Lemma 1.3. The element [X, X] is nonzero.

Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group

### 4.4 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)

### Algebraic 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

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

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

### SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS

SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e in a general complete intersection of multidegree

### Characteristic classes in the Chow ring

arxiv:alg-geom/9412008v1 10 Dec 1994 Characteristic classes in the Chow ring Dan Edidin and William Graham Department of Mathematics University of Chicago Chicago IL 60637 Let G be a reductive algebraic

### 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

### 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)

### Integral Extensions. Chapter Integral Elements Definitions and Comments Lemma

Chapter 2 Integral Extensions 2.1 Integral Elements 2.1.1 Definitions and Comments Let R be a subring of the ring S, and let α S. We say that α is integral over R if α isarootofamonic polynomial with coefficients

### SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS

SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e on a general complete intersection of multidegree

### Algebraic 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......................

### Math 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)

### Homogeneous Coordinate Ring

Students: Kaiserslautern University Algebraic Group June 14, 2013 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4 Outline Quotients in Algebraic Geometry 1 Quotients in

### HYPERSURFACES IN PROJECTIVE SCHEMES AND A MOVING LEMMA

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

### 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

### monomialization of strongly prepared morphisms from nonsingular n-folds to surfaces

monomialization of strongly prepared morphisms from nonsingular n-folds to surfaces Steven D Cutosy and Olga Kashcheyeva Introduction Monomialization of morphisms is the problem of transforming a mapping

### LINKED HOM SPACES BRIAN OSSERMAN

LINKED HOM SPACES BRIAN OSSERMAN Abstract. In this note, we describe a theory of linked Hom spaces which complements that of linked Grassmannians. Given two chains of vector bundles linked by maps in both

### Exploring 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

### Basic facts and definitions

Synopsis Thursday, September 27 Basic facts and definitions We have one one hand ideals I in the polynomial ring k[x 1,... x n ] and subsets V of k n. There is a natural correspondence. I V (I) = {(k 1,

### Math 396. Bijectivity vs. isomorphism

Math 396. Bijectivity vs. isomorphism 1. Motivation Let f : X Y be a C p map between two C p -premanifolds with corners, with 1 p. Assuming f is bijective, we would like a criterion to tell us that f 1

### 12 Hilbert polynomials

12 Hilbert polynomials 12.1 Calibration Let X P n be a (not necessarily irreducible) closed algebraic subset. In this section, we ll look at a device which measures the way X sits inside P n. Throughout

### 3 Lecture 3: Spectral spaces and constructible sets

3 Lecture 3: Spectral spaces and constructible sets 3.1 Introduction We want to analyze quasi-compactness properties of the valuation spectrum of a commutative ring, and to do so a digression on constructible

### We simply compute: for v = x i e i, bilinearity of B implies that Q B (v) = B(v, v) is given by xi x j B(e i, e j ) =

Math 395. Quadratic spaces over R 1. Algebraic preliminaries Let V be a vector space over a field F. Recall that a quadratic form on V is a map Q : V F such that Q(cv) = c 2 Q(v) for all v V and c F, and

### Exercise 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

### GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS

GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS ANA BALIBANU DISCUSSED WITH PROFESSOR VICTOR GINZBURG 1. Introduction The aim of this paper is to explore the geometry of a Lie algebra g through the action

### 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.

### Yuriy 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

### div(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.

### The 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 ]

### Commutative Algebra. Andreas Gathmann. Class Notes TU Kaiserslautern 2013/14

Commutative Algebra Andreas Gathmann Class Notes TU Kaiserslautern 2013/14 Contents 0. Introduction......................... 3 1. Ideals........................... 9 2. Prime and Maximal Ideals.....................

### MATH 326: RINGS AND MODULES STEFAN GILLE

MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called

### Introduction to Arithmetic Geometry Fall 2013 Lecture #15 10/29/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #15 10/29/2013 As usual, k is a perfect field and k is a fixed algebraic closure of k. Recall that an affine (resp. projective) variety is an

### 2. 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.

### Math 216A. A gluing construction of Proj(S)

Math 216A. A gluing construction o Proj(S) 1. Some basic deinitions Let S = n 0 S n be an N-graded ring (we ollows French terminology here, even though outside o France it is commonly accepted that N does

### 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

### Math 676. A compactness theorem for the idele group. and by the product formula it lies in the kernel (A K )1 of the continuous idelic norm

Math 676. A compactness theorem for the idele group 1. Introduction Let K be a global field, so K is naturally a discrete subgroup of the idele group A K and by the product formula it lies in the kernel

### 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.

### LEVINE S CHOW S MOVING LEMMA

LEVINE S CHOW S MOVING LEMMA DENIS NARDIN The main result of this note is proven in [4], using results from [2]. What is here is essentially a simplified version of the proof (at the expense of some generality).

### Rings With Topologies Induced by Spaces of Functions

Rings With Topologies Induced by Spaces of Functions Răzvan Gelca April 7, 2006 Abstract: By considering topologies on Noetherian rings that carry the properties of those induced by spaces of functions,

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

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

### A SHORT PROOF OF ROST NILPOTENCE VIA REFINED CORRESPONDENCES

A SHORT PROOF OF ROST NILPOTENCE VIA REFINED CORRESPONDENCES PATRICK BROSNAN Abstract. I generalize the standard notion of the composition g f of correspondences f : X Y and g : Y Z to the case that X

### This 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

### ALGEBRAIC HYPERBOLICITY OF THE VERY GENERAL QUINTIC SURFACE IN P 3

ALGEBRAIC HYPERBOLICITY OF THE VERY GENERAL QUINTIC SURFACE IN P 3 IZZET COSKUN AND ERIC RIEDL Abstract. We prove that a curve of degree dk on a very general surface of degree d 5 in P 3 has geometric

### Spring 2016, lecture notes by Maksym Fedorchuk 51

Spring 2016, lecture notes by Maksym Fedorchuk 51 10.2. Problem Set 2 Solution Problem. Prove the following statements. (1) The nilradical of a ring R is the intersection of all prime ideals of R. (2)

### 11. 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

### Lemma 1.3. The dimension of I d is dimv d D = ( d D+2

BEZOUT THEOREM One of the most fundamental results about the degrees of polynomial surfaces is the Bezout theorem, which bounds the size of the intersection of polynomial surfaces. The simplest version

### Parameterizing orbits in flag varieties

Parameterizing orbits in flag varieties W. Ethan Duckworth April 2008 Abstract In this document we parameterize the orbits of certain groups acting on partial flag varieties with finitely many orbits.

### INTRODUCTION TO ALGEBRAIC GEOMETRY. Throughout these notes all rings will be commutative with identity. k will be an algebraically

INTRODUCTION TO ALGEBRAIC GEOMETRY STEVEN DALE CUTKOSKY Throughout these notes all rings will be commutative with identity. k will be an algebraically closed field. 1. Preliminaries on Ring Homomorphisms

### NONSINGULAR CURVES BRIAN OSSERMAN

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

### Honors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35

Honors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35 1. Let R 0 be a commutative ring with 1 and let S R be the subset of nonzero elements which are not zero divisors. (a)

### ALGORITHMS FOR ALGEBRAIC CURVES

ALGORITHMS FOR ALGEBRAIC CURVES SUMMARY OF LECTURE 7 I consider the problem of computing in Pic 0 (X) where X is a curve (absolutely integral, projective, smooth) over a field K. Typically K is a finite

### 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,

### ADVANCED 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

### MATH32062 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,

### TWO IDEAS FROM INTERSECTION THEORY

TWO IDEAS FROM INTERSECTION THEORY ALEX PETROV This is an expository paper based on Serre s Local Algebra (denoted throughout by [Ser]). The goal is to describe simple cases of two powerful ideas in intersection

### 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