# COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY

Size: px
Start display at page:

Download "COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY"

Transcription

1 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 topology and its many pathologies, because they already had a better-behaved topology to work with: the analytic topology inherited from the usual topology on the complex numbers themselves. In this note, we introduce the analytic topology, and explore some of its basic properties. We also investigate how it interacts with properties of varieties which we have already defined. We work throughout in the context of schemes of finite type over a field (generally the complex numbers). However, our considerations will for the most part be topological, so any non-reduced structures will not be relevant to us. If X is of finite type over a field k, we denote by X(k) the set of points of X with residue field k. [It is sometimes useful to recall that this is the same as the set of morphisms Spec k X over Spec k.] Our convention is that a variety over k is an integral separated scheme of finite type over k, and a curve is a variety of dimension 1 (in particular, irreducible). The proofs are largely adapted from [5] and [4]. 1. The analytic topology on affine schemes If X A n C is a closed subscheme, we will endow X(C) with a topology which corresponds far more closely than the Zariski topology to our intuition for what X looks like. We define: Definition 1.1. The analytic topology on X is the topology induced by the inclusion X(C) A n C (C) = Cn, using the usual topology on C n. The topological space of X(C) endowed with the analytic topology is denoted by X an. Because zero sets of (multivariate) polynomials are closed in C n, the analytic topology is finer than the Zariski topology: that is, a closed subset in the Zariski topology is closed in the analytic topology, but not in general vice versa. This may be rephrased into the following conclusion: Proposition 1.2. The map of topological spaces X an X induced by the identity on points is continuous. Similar arguments also prove the following basic facts. Exercise 1.3. (a) If also Y A m C is a closed subscheme, and X Y is a C-morphism, then the induced map X an Y an is continuous. In particular, a section of O X (X) induces a continuous map X an C = (A 1 C ) an. (b) The analytic topology on X is an isomorphism invariant, independent of the particular imbedding of X into affine space. (c) If Z X is a subscheme, then Z an = (X an ) Z(C). We will need one elementary result on the continuity of roots of a single-variable complex polynomial. Theorem 1.4. Let f(x) = a 0 + a 1 x + + a d x d be a nonzero complex polynomial, and let c C be a root of f. Then for any ɛ > 0, there exists δ > 0 such that for any b 0,..., b d C with a i b i < δ for all i, there is a root c of the polynomial g(x) = b 0 + b 1 x + + b d x d with c c < ɛ. 1

2 Proof. Let γ be a circle of radius less than ɛ around c, chosen so that there are no other zeros of f(x) on γ. Let w be the minimum absolute value of f(x) on γ, which is strictly positive by construction. For δ sufficiently small, we have that if b 0,..., b d C satisfy a i b i < δ, then f(x) g(x) < w for all x γ, where g(x) = b 0 + b 1 x + + b d x d. It follows from Rouche s theorem that f and g have the same number of roots inside the circle γ, which gives the desired statement. We can now begin to make statements about the analytic topology of varieties, at least in some special cases. Corollary 1.5. Let C A 2 C be a curve in the plane. Then C an has no isolated points. Proof. Since C has codimension 1 in A 2 C, we know it can be expressed as the zero set of a single polynomial, say f(x, y) C[x, y], with deg f = d. Then f(x, y) has degree at most d when considered as a polynomial in y, and its coefficients are themselves continuous functions of x. We may assume that f(x, y) is not constant in y, since otherwise by irreducibility of C we must have f = x c for some c, so C is a vertical line and certainly has no isolated points. Thus, given (x 0, y 0 ) C, it follows from Theorem 1.4 that for any ɛ > 0, there is some δ > 0 such that for every x with x x 0 < δ, there is some y with y y 0 < ɛ and f(x, y) = 0. We conclude that (x 0, y 0 ) is not an isolated point of C. Example 1.6. To see that the above Corollary has some content to it, we observe that it is false over the real numbers. Indeed, the curve y 2 = x 3 x 2 is irreducible over R, but its real points consist of a single one-dimensional component supported on x 1, and a single isolated point at the origin. 2. The analytic topology on schemes Having defined the analytic topology on affine schemes, we can now define it on arbitrary schemes, via gluing. Definition 2.1. Let X be a scheme of finite type over C. The analytic topology on X, denoted by X an, is the topology such that for each affine open subset U, the natural map U an (X an ) U(C) is a homeomorphism. Once again, there are some basic properties to check for the analytic topology: Exercise 2.2. (a) The analytic topology X an on a prevariety X is well defined. (b) The map of topological spaces X an X induced by the identity on points is continuous. (c) Given another scheme Y of finite type over C, and a C-morphism ϕ : X Y, the induced map X an Y an is continuous. (d) If Z X is a subscheme, then Z an = (X an ) Z(C). Example 2.3. Complex projective space P n C (which is just the classical CPn ) is compact in the analytic topology. Indeed, we have the natural morphism A n+1 C {0} P n C, which by Exercise 2.2 (c) induces a continuous map (A n+1 C {0}) an (P n C ) an. Inside (A n+1 C {0}) an we have the closed subset consisting of elements of norm 1; identifying A n+1 C with R 2n+2, this subset is precisely the unit sphere, and hence compact. Moreover, it surjects onto (P n C ) an, and since the continuous image of a compact set is compact, we conclude that (P n C ) an is compact. As an immediate consequence of Example 2.3 and Exercise 2.2, we conclude: Corollary 2.4. If X is a projective scheme over C, then X an is compact. We know that if X, Y are schemes over C, the Zariski topology on X Spec C Y is not the product topology. However, (again confirming that the analytic topology behaves closer to our intuition) for the analytic topology we have: 2

3 Exercise 2.5. If X, Y are schemes of finite type over C, then (X Spec C Y ) an = X an ) (Y an, where on the righthand side we take the product topology. Note that by the universal property of fibered products, we at least have (X Spec C Y )(C) canonically identified with X(C) Y (C). We can now conclude that separatedness implies that the analytic topology is Hausdorff. Corollary 2.6. If X is a separated scheme of finite type over C, then X an is Hausdorff. Proof. For X an to be Hausdorff is equivalent to the image (X an ) in X an X an to be closed. Since X is separated, (X) is closed in X X in the Zariski topology, so (X an ) is closed in (X X) an by Exercise 2.2 (b). But (X X) an = X an X an by Exercise 2.5, so we conclude that X an is Hausdorff. In fact, we will see in Corollary 3.3 that the converse also holds: if X is a complex prevariety and X an is Hausdorff, then X is a variety. However, this requires putting together some of the deeper results which we have developed. We can generalize Corollary 1.5 as follows: Corollary 2.7. Let X be a complex curve, or more generally an irreducible one-dimensional scheme of finite type over C. Then X an has no isolated points. Before we give the proof, we note that the analytic topology on a prevariety X is locally metrizable (and indeed metrizable if X is separated, from Corollary 2.6). Thus we can work more conceptually with sequential limits and compactness. Proof. We prove the statement in several steps. We first prove it for projective nonsingular curves X P n C. Given P X, we know that there exists a morphism ϕ : X P2 C such that ϕ 1 (ϕ(p )) = {P }. Since ϕ(x) is a (projective) plane curve, noting that we can replace it by an affine open neighborhood of ϕ(p ), we know by Corollary 1.5 that ϕ(p ) is not an isolated point of ϕ(x) an. Now, let Q 1, Q 2,... be a sequence of points of ϕ(x) approaching ϕ(p ) in the analytic topology, and lift them to P 1, P 2,... in X. We know that X an is compact by Corollary 2.4, so some subsequence of P 1, P 2,... converges to a point P X in the analytic topology. But then since ϕ is continuous, we conclude ϕ(p ) = ϕ(p ), so P = P, and P is not an isolated point of X. The case of an arbitrary nonsingular curve X then follows, since we know that X can be realized as a Zariski open subset of some nonsingular projective X, and X X consists of finitely many points, so if Xan has no isolated points then X an cannot have any isolated points either. Next, we conclude the statement of the corollary for an arbitrary curve X by considering the normalization morphism ν : X X; this is a surjective morphism, with X a nonsingular curve. For any P X, we have ν 1 (P ) a closed set, hence a finite set of points, and since we know that X an has no isolated points, if we choose a sequence in X converging to a point of ν 1 (P ), it contains at most finitely many points in ν 1 (P ), and its image is then a sequence converging to P, so X an does not have any isolated points. Finally, reducedness is irrelevant since the question is topological, and separatedness is likewise not an issue, as an isolated point would remain isolated in an affine (hence separated) open neighborhood, so we conclude the more general statement as well. 3. Separatedness and properness There are a number of basic and important facts relating the ideas we have introduced for schemes to standard topological properties applied to the analytic topology. The main ingredient for proving these statements is the following: 3

4 Theorem 3.1. Let X be a scheme of finite type over C, and U a Zariski-dense open subset. Then U an is dense in X an. Proof. We first observe that the statement of the theorem in the case that X is a curve is precisely Corollary 2.7. Now, for arbitrary X, given P X U, by Lemma 3.1 of Properties of fibers and applications there exists a one-dimensional integral closed subscheme Z of X with P Z and Z U. Restricting Z to an open neighborhood of P, we may assume Z is separated, hence a curve. By Exercise 2.2 (d), we have Z an = (X an ) Z(C), and P is in the (analytic) closure of Z an U since we already that the result hold for curves. It thus follows that P is in the closure of U in X an, and we conclude that U is dense, as asserted. In particular, we conclude the following: Corollary 3.2. Let X be a scheme of finite type over C, and Z X a subscheme. If Z an is closed in X an, then Z is a closed subscheme of X. Proof. It is enough to show that Z is closed in X in the Zariski topology. The question being topological, we may assume that Z is reduced, and then if Z denotes the closure in X, we have from the definition of subscheme that Z is Zariski open (and dense) in Z, so by Theorem 3.1, we conclude that Z an is dense in Z an. Then if Z an is closed in X an, we conclude that Z an = Z an, and hence Z(C) = Z(C). Now, (Z) Z is of finite type over C, so if nonempty, it would have to contain a point of Z(C). We thus conclude that Z = Z, as desired. We can now prove several basic statements on the analytic topology. Corollary 3.3. Let X be of finite type over C. Then X is separated over C if and only if X an is Hausdorff. Proof. One direction was proved already in Corollary 2.6. Conversely, suppose X an is Hausdorff, so that the diagonal (X) is closed in the analytic topology. Then by Corollary 3.2, we have (X) closed also in the Zariski topology, so X is separated. To apply Theorem 3.1 to study compactness, we will want a more general form of Chow s lemma than is stated in [2]. Lemma 3.4. Let S be a Noetherian scheme, and X separated and of finite type over S. Then there exists a scheme X and a surjective projective morphism f : X X such that X is quasiprojective over S, and there is a dense open subset U X such that f U : f 1 (U) U is an isomorphism. The proof is essentially the same; see Theorem of [1]. Note that the form in [2] follows from this, since if X is proper over S, then X is both quasi-projective and proper over S, and it follows that X is projective over S. Corollary 3.5. A separated scheme X of finite type over C is proper over C if and only if X an is compact. Proof. First suppose that X an is compact, and X is quasiprojective. Then given an immersion X P n C, we have that the image of X an is closed in (P n C ) an, so by Corollary 3.2 we have that X is a closed subscheme, hence projective. In the general case, let f : X X be as in Chow s lemma. Since f is projective and X an is compact, we conclude that (X ) an is compact, hence projective over C. Finally, because surjectivity is closed under base change, we see easily that since X is universally closed over C, we must have X universally closed as well, hence proper. For the converse, first suppose that X is projective. Then X an is compact by Corollary 2.4. Now suppose instead that X is proper. Then again using Lemma 3.4, there is a surjective morphism X X for some projective variety X. Thus, X an is the continuous image of the compact space (X ) an, and is therefore compact. 4

5 4. A digression on nonsingularity In order to conclude our study of the complex topology, we will need to know an important and basic fact: a nonsingular complex curve has the natural structure of a (one-dimensional) complex manifold. We will prove the more general statement for complex varieties of any dimension. Before doing so, we need to develop a few basic ideas that apply over any field. Definition 4.1. Let X be a scheme of finite type over a field k, and x X a nonsingular point, at which dim X = d. A system of local coordinates for X and x is a d-tuple (t 1,..., t d ) of elements of the maximal ideal m x of O X,x which gives a basis for the Zariski cotangent space m x /m 2 x. Note that by Nakayama s lemma, (t 1,..., t d ) is a system of local coordinates if and only if the t i generate the maximal ideal. We will need the Jacobian criterion for nonsingularity, which we state in a slightly special case below: Theorem 4.2. Let X A n k be an affine scheme given by an ideal I k[t 1,..., t n ]. Given x X(k), suppose that dim O X,x d. Let c 1,..., c n k be the values of the t i at x. Then the following are equivalent: (1) X is nonsingular at x of dimension d, and (t n d+1 c n d+1,..., t n c n ) induces a system of local coordinates for X at x; ( (2) there exist f 1,..., f n d I such that the Jacobian matrix fi t j is invertible at x. )1 i,j n d Furthermore, in this case we have that f 1,..., f n d generates I in a neighborhood of x. Proof. Denote by Tx X and Tx A n k the Zariski cotangent spaces at x of X and An k, respectively (that is, the maximal ideal modulo its square in the local rings at x). Given f k[t 1,..., t n ], denote by df x the element of Tx A n k induced by f f(c 1,..., c n ). Since κ(x) = k, we have that both Tx X and Tx A n k spaces are k-vector spaces, with the latter of dimension n and having basis given by the d(t i ) x. We also have the natural surjection π : Tx A n k T x X, whose kernel we observe is precisely the image of I in T A n k. Furthermore, we know that dim T x X d, with equality if and only if X is nonsingular of dimension d at x. Thus, we obtain the following equivalences: (1) holds; the images under π of d(t n d+1 ) x,..., d(t n ) x span the space Tx X; the kernel of π has dimension n d, and is disjoint from the span of d(t n d+1 ) x,..., d(t n ) x ; the kernel of π maps isomorphically onto the span of d(t 1 ) x,..., d(t n d ) x under projection to the first n d coordinates; the kernel of π surjects onto the span of d(t 1 ) x,..., d(t n d ) x under projection to the first n d coordinates; there exist f 1,..., f n d I such that the span of the (df i ) x surjects onto the span of d(t 1 ) x,..., d(t n d ) x under projection to the first n d coordinates. We then observe that given any f 1,..., f nd vanishing at x, the map k n d k n d given by projecting each (df i ) x to the span of d(t 1 ) x,..., d(t n d ) x is given by the Jacobian matrix of condition (2). Thus, we conclude that the final condition above is equivalent to (2), as desired. For the last assertion, we cheat a bit, insofar as we will invoke the fact (which we haven t proved) that regular local rings are integral domains. Let J = (f 1,..., f n d ), and set Y A n k to be the closed subscheme on which J vanishes, so that X is a closed subscheme of Y. Then by the Krull principal ideal theorem, Y has dimension at least d at all its closed points, and in particular at x. Applying the equivalence of (1) and (2), we conclude that Y is nonsingular of dimension d at x. Now we use that both O X,x and O Y,x are integral domains: the generic point of X must correspond to a prime ideal of O Y,x, but in a neighborhood of x, but we have that dim O X,x = dim O Y,x, so the generic point of X must correspond to the zero ideal in O Y,x, so O X,x = O Y,x. We thus have 5

6 that f 1,..., f n d generate I in O A n k,x. We finally conclude (using that I is finitely generated) that the f i generate I in a neighborhood of x. Remark 4.3. In addition to standard commutative algebra citations for regular local rings being integral domains (for instance, Theorem 14.3 of [3]), one can prove the result in the case we need by developing the theory of completions and power series; see Theorem 3.1 of the notes Power series and nonsingular points from Math 248B, Winter Just for fun, we mention the following corollary, which says that nonsingular points are locally complete intersections : Corollary 4.4. Let X be of finite type over k, and x X(k) a nonsingular point of X of dimension d. Then for any neighborhood U of x, and any closed immersion U A n k with U defined by an ideal I, there exists f 1,..., f n d I such that the f i generate I in an open neighborhood of x. Proof. If A n k has coordinates t 1,..., t n, and in these coordinates x = (c 1,..., c n ), then since the natural map O A n k,x O X,x is a surjection, we have that the t i c i generate the maximal ideal of O X,x, and some d of them must give a system of local coordinates for X at x. Without loss of generality, we may assume these are t n d+1 c n d+1,..., t n c n. We then conclude the desired statement from Theorem 4.2. We now return to the complex case with the following corollary of Theorem 4.2: Corollary 4.5. Let X be a scheme of finite type over C, and x X(C) a nonsingular point of X. If (t 1,..., t d ) is a system of local coordinates for X at x, there exists a neighborhood U of x in X an on which all the t i induce sections of O X, and such that the induced map U C d defines a homeomorphism of U an onto an open neighborhood V of the origin in C d, with every element of O X corresponding to an analytic function on the image of its domain of definition in V. In particular, if X is a nonsingular variety of dimension d, then X an has the structure of a complex manifold of dimension d, with sections of O X giving rise to analytic functions on X an. Proof. The first statement being local on X, we may assume that X = Spec A is affine and the t i are global sections of O X. Since X is of finite type, for some n we can extend the t i to a set of n generators of A over C, and let I be the corresponding ideal of definition for X in A n C. Reindexing and applying Theorem 4.2, we conclude that I is generated in a neighborhood of x by some f 1,..., f n d, and that ( fi t j )1 i,j n d is invertible at x. By the implicit function theorem, there exist neighborhoods U of x in X an, V of the origin in C d and an analytic function g : V C n d such that id g : V C n maps onto U and is inverse to the projection map C n C d. This gives the desired homeomorphism. In addition, g expresses all the t i as analytic functions of t 1,..., t d, and since sections of O X are expressed locally as rational functions in the t i with denominators nonvanishing on their domain of definition, we conclude that they are analytic on their domain of definitions. For the final statement, if X is a variety it is in particular separated, so X an is Hausdorff by Corollary 2.6. The first statement gives us a complex atlas on an open cover of X an, and to conclude that the induced transition maps are analytic, we need only use that algebraic isomorphisms are analytic under above correspondence, which follows easily from the fact that sections of O X yield analytic functions. 5. A digression on the Riemann-Roch theorem We will need to know the following fact: Theorem 5.1. Let X be a nonsingular projective curve over a field k, and x X a closed point. Then there exists a (nonconstant) rational function f K(X) with a pole only at x. 6

7 This is an immediate consequence of the Riemann-Roch theorem; see for instance Theorem IV.1.3 of [2]. However, if one only wants to prove this type of statement, it suffices to prove an inequality, which is easier. See III.6.6, Lemma of [5]. Remark 5.2. A stronger but closely related version of Theorem 5.1 is that if X is a nonsingular projective curve over k, and x X a closed point, then there exists a closed immersion of X into P n k for some n, such that there exists a hyperplane H which intersects the image of X only in the image of x. One can show that an open subset of an affine nonsingular curve is still affine, and it follows from this and the above that any nonsingular curve which is not projective is in fact affine. 6. Connectedness We conclude our discussion of the complex topology with the following foundational theorem: Theorem 6.1. Let X be of finite type over C. Then X is connected in the Zariski topology if and only if X an is connected. As we shall see, this theorem is quite a bit deeper than the corresponding statements for properness and separatedness. This difficulty should perhaps not be surprising: if we consider the analogous statement over R, it remains true that a separated scheme is Hausdorff in the real analytic topology, and a proper scheme is compact, but it is not true that a (Zariski) connected scheme is connected in the real analytic topology, as we see already with elliptic curves in the plane. Nonetheless, with the tools we have developed the proof of the theorem will not be very difficult. Proof. Since the analytic topology is finer than the Zariski topology, it is clear that if X an is connected, then so is X. For the converse, we begin by proving that the desired result holds in the case that X is a nonsingular projective curve. Given x X, we know from Theorem 5.1 that there exists a (nonconstant) rational function f with a pole only at x. We know by Corollary 3.5 and Corollary 4.5 that X an is a compact one-dimensional complex manifold, and f (being a quotient of analytic functions) induces a meromorphic function on X an. If X an is disconnected, let C X an be a connected component not containing x. Then f is analytic on C, so the maximum modulus principle implies that f is constant on C. But then subtracting this constant we obtain a non-zero rational function with infinitely many zeroes on X, which is impossible. Thus, X an is connected. Next, if X is an arbitrary nonsingular curve, we know that it may be imbedded as an open subset of a nonsingular projective curve X. We now know that X an is a connected complex manifold of dimension 1. But the complement of X an in X an is a finite set of points, so we conclude that X an is likewise connected, as desired. Now suppose that X is any curve, and let X X be the normalization. Then we have shown that X an connected, but X an surjects onto X an, so we conclude that X an is likewise connected. Finally, suppose that X is arbitrary of dimension 1. Inducting on the number of components, we reduce to the case that X is irreducible. Passing to an affine open subset, we omit only finitely many points, which by Corollary 2.7 cannot be isolated, so connnectedness is not affected. We may thus assume that X is separated, and since reducedness is irrelevant, we have reduced to the previously addressed case that X is a curve. To address the case of arbitrary dimension, we have proved in Theorem 3.2 of Properties of fibers and applications that any two points of X(C) lie in some connected 1-dimensional closed subscheme Z. Since we now know that Z an is connected, we conclude that X an is connected, as desired. 7

8 References 1. Alexander Grothendieck and Jean Dieudonné, Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes, Publications mathématiques de l I.H.É.S., vol. 8, Institut des Hautes Études Scientifiques, Robin Hartshorne, Algebraic geometry, Springer-Verlag, Hideyuki Matsumura, Commutative ring theory, Cambridge University Press, David Mumford, The red book of varieties and schemes, second, expanded ed., Springer, Igor Shafarevich, Basic algebraic geometry 1. varieties in projective space, second ed., Springer-Verlag,

### 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

### CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES

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

### 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

### 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

### DEFORMATIONS VIA DIMENSION THEORY

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

### 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

### NOTES ON DIMENSION THEORY OF SCHEMES

NOTES ON DIMENSION THEORY OF SCHEMES BRIAN OSSERMAN In this expository note, we discuss various aspects of the theory of dimension of schemes, in particular focusing on which hypotheses are necessary in

### SEPARATED AND PROPER MORPHISMS

SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Last quarter, we introduced the closed diagonal condition or a prevariety to be a prevariety, and the universally closed condition or a variety to be complete.

### VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS

VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Recall that or prevarieties, we had criteria or being a variety or or being complete in terms o existence and uniqueness o limits, where

### SEPARATED AND PROPER MORPHISMS

SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN The notions o separatedness and properness are the algebraic geometry analogues o the Hausdor condition and compactness in topology. For varieties over the

### VALUATIVE CRITERIA BRIAN OSSERMAN

VALUATIVE CRITERIA BRIAN OSSERMAN Intuitively, one can think o separatedness as (a relative version o) uniqueness o limits, and properness as (a relative version o) existence o (unique) limits. It is not

### 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

### 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

### 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

### (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

### 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

### 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

### 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

### INTRODUCTION 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:

### 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

### 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27

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

### 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)

### AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES

AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES DOMINIC L. WYNTER Abstract. We introduce the concepts of divisors on nonsingular irreducible projective algebraic curves, the genus of such a curve,

### 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

### PROBLEMS, 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

### 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,

### 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

### 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

### 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 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

### 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.

### 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

### 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,

### 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

### Course 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...............................

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

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

### 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).

### MATH 8254 ALGEBRAIC GEOMETRY HOMEWORK 1

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

### Lecture 7: Etale Fundamental Group - Examples

Lecture 7: Etale Fundamental Group - Examples October 15, 2014 In this lecture our only goal is to give lots of examples of etale fundamental groups so that the reader gets some feel for them. Some of

### SCHEMES. David Harari. Tsinghua, February-March 2005

SCHEMES David Harari Tsinghua, February-March 2005 Contents 1. Basic notions on schemes 2 1.1. First definitions and examples.................. 2 1.2. Morphisms of schemes : first properties.............

### Math 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

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

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

### 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

### Topological 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

### 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.

### I(p)/I(p) 2 m p /m 2 p

Math 6130 Notes. Fall 2002. 10. Non-singular Varieties. In 9 we produced a canonical normalization map Φ : X Y given a variety Y and a finite field extension C(Y ) K. If we forget about Y and only consider

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

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

### Introduction 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

### Algebraic 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

### Algebraic 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

### Math General Topology Fall 2012 Homework 8 Solutions

Math 535 - General Topology Fall 2012 Homework 8 Solutions Problem 1. (Willard Exercise 19B.1) Show that the one-point compactification of R n is homeomorphic to the n-dimensional sphere S n. Note that

### 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

### Dedekind Domains. Mathematics 601

Dedekind Domains Mathematics 601 In this note we prove several facts about Dedekind domains that we will use in the course of proving the Riemann-Roch theorem. The main theorem shows that if K/F is a finite

### 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

### 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

### INTRODUCTION TO REAL ANALYTIC GEOMETRY

INTRODUCTION TO REAL ANALYTIC GEOMETRY KRZYSZTOF KURDYKA 1. Analytic functions in several variables 1.1. Summable families. Let (E, ) be a normed space over the field R or C, dim E

### 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,

### ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF

ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF MATTHEW H. BAKER AND JÁNOS A. CSIRIK Abstract. We give a new proof of the isomorphism between the dualizing sheaf and the canonical

### B 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

### Math 210B. Artin Rees and completions

Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show

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

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

### 1. 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.

### GLUING 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

### 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 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 18

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 18 CONTENTS 1. Invertible sheaves and divisors 1 2. Morphisms of schemes 6 3. Ringed spaces and their morphisms 6 4. Definition of morphisms of schemes 7 Last day:

### Riemann Surfaces and Algebraic Curves

Riemann Surfaces and Algebraic Curves JWR Tuesday December 11, 2001, 9:03 AM We describe the relation between algebraic curves and Riemann surfaces. An elementary reference for this material is [1]. 1

### 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

### 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

### Infinite-Dimensional Triangularization

Infinite-Dimensional Triangularization Zachary Mesyan March 11, 2018 Abstract The goal of this paper is to generalize the theory of triangularizing matrices to linear transformations of an arbitrary vector

### 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

### MATH 631: ALGEBRAIC GEOMETRY: HOMEWORK 1 SOLUTIONS

MATH 63: ALGEBRAIC GEOMETRY: HOMEWORK SOLUTIONS Problem. (a.) The (t + ) (t + ) minors m (A),..., m k (A) of an n m matrix A are polynomials in the entries of A, and m i (A) = 0 for all i =,..., k if and

### 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

### 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

### 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......................

### 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

### ALGEBRAIC 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

### DIFFERENTIAL TOPOLOGY AND THE POINCARÉ-HOPF THEOREM

DIFFERENTIAL TOPOLOGY AND THE POINCARÉ-HOPF THEOREM ARIEL HAFFTKA 1. Introduction In this paper we approach the topology of smooth manifolds using differential tools, as opposed to algebraic ones such

### 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.

### MODULI SPACES OF CURVES

MODULI SPACES OF CURVES SCOTT NOLLET Abstract. My goal is to introduce vocabulary and present examples that will help graduate students to better follow lectures at TAGS 2018. Assuming some background

### 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

### 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

### 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

### 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

### INTERSECTION 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

### 1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

### Lecture 6: Etale Fundamental Group

Lecture 6: Etale Fundamental Group October 5, 2014 1 Review of the topological fundamental group and covering spaces 1.1 Topological fundamental group Suppose X is a path-connected topological space, and

### A ne Algebraic Varieties Undergraduate Seminars: Toric Varieties

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

### MOISHEZON SPACES IN RIGID GEOMETRY

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

### 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

### 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 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)

### INVERSE LIMITS AND PROFINITE GROUPS

INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological

### Néron Models of Elliptic Curves.

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

### 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

### (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

### 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...............................