Grothendieck Ring of Varieties
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1 Affine Varieties
1 Affine Varieties We will begin following Kempf's Algebraic Varieties, and eventually will do things more like in Hartshorne. We will also use various sources for commutative algebra. What is algebraic geometry? Classically, it is the study of the zero sets of polynomials. We will now fix some notation. k will be some fixed algebraically closed field, any ring is commutative with identity, ring homomorphisms preserve identity, and a k-algebra is a ring R which contains k (i.e., we have a ring homomorphism ι : k ! R). P ⊆ R an ideal is prime iff R=P is an integral domain. Algebraic Sets n n We define affine n-space, A = k = f(a1; : : : ; an): ai 2 kg. n Any f = f(x1; : : : ; xn) 2 k[x1; : : : ; xn] defines a function f : A ! k : (a1; : : : ; an) 7! f(a1; : : : ; an). Exercise If f; g 2 k[x1; : : : ; xn] define the same function then f = g as polynomials. Definition 1.1 (Algebraic Sets). Let S ⊆ k[x1; : : : ; xn] be any subset. Then V (S) = fa 2 An : f(a) = 0 for all f 2 Sg. A subset of An is called algebraic if it is of this form. e.g., a point f(a1; : : : ; an)g = V (x1 − a1; : : : ; xn − an). Exercises 1. I = (S) is the ideal generated by S. Then V (S) = V (I). 2. I ⊆ J ) V (J) ⊆ V (I). P 3. V ([αIα) = V ( Iα) = \V (Iα). 4. V (I \ J) = V (I · J) = V (I) [ V (J). Definition 1.2 (Zariski Topology). We can define a topology on An by defining the closed subsets to be the algebraic subsets. -
NOTES on CARTIER and WEIL DIVISORS Recall: Definition 0.1. A
NOTES ON CARTIER AND WEIL DIVISORS AKHIL MATHEW Abstract. These are notes on divisors from Ravi Vakil's book [2] on scheme theory that I prepared for the Foundations of Algebraic Geometry seminar at Harvard. Most of it is a rewrite of chapter 15 in Vakil's book, and the originality of these notes lies in the mistakes. I learned some of this from [1] though. Recall: Definition 0.1. A line bundle on a ringed space X (e.g. a scheme) is a locally free sheaf of rank one. The group of isomorphism classes of line bundles is called the Picard group and is denoted Pic(X). Here is a standard source of line bundles. 1. The twisting sheaf 1.1. Twisting in general. Let R be a graded ring, R = R0 ⊕ R1 ⊕ ::: . We have discussed the construction of the scheme ProjR. Let us now briefly explain the following additional construction (which will be covered in more detail tomorrow). L Let M = Mn be a graded R-module. Definition 1.1. We define the sheaf Mf on ProjR as follows. On the basic open set D(f) = SpecR(f) ⊂ ProjR, we consider the sheaf associated to the R(f)-module M(f). It can be checked easily that these sheaves glue on D(f) \ D(g) = D(fg) and become a quasi-coherent sheaf Mf on ProjR. Clearly, the association M ! Mf is a functor from graded R-modules to quasi- coherent sheaves on ProjR. (For R reasonable, it is in fact essentially an equiva- lence, though we shall not need this.) We now set a bit of notation. -
3. Topological K-Theory Shortly After Grothendieck Introduced K0(X)
3. Topological K-theory Shortly after Grothendieck introduced K0(X) for an algebraic variety X, M. Atiyah and F. Hirzebrch introduced the analogous theory for topological spaces ∗ [2] based on topological vector bundles. We consider this theory Ktop(−) for two reasons: first, the theory developed by Atiyah and Hirzebruch has been a model for 40 years of effort in algebraic K-theory, effort that has recently produced significant advances; second, topological K-theory of the underlying analytic space of a com- plex variety X, Xan, provides a much more computable theory to which algebraic K-theory maps. A (complex) topological vector bundle of rank r, p : E → T , on a space T is a continuous map with fibers each given the structure of a complex vector space of dimension r such that there is an open covering {Ui ⊂ T } with the property that r ∼ there exist homeomorphisms φi : C ×Ui → E|Ui over Ui which are C-linear on each fiber. One readily verifies that such a topological vector bundle p : E → T , on T of rank r determines and is determined by patching data: a collection of continuous, fiber-wise linear homemorphisms for each i, j −1 r r φj ◦ φi = θi,j : C × (Ui ∩ Uj) ' (E|Ui )Uj = (E|Uj )Ui ' C × (Ui ∩ Uj). This in turn is equivalent to a 1-cocycle 1 {θi,j ∈ Mapscont(Ui,j, GL(r, C))} ∈ Z (T, GL(r, C)). Indeed, two such 1-cocycles determine isomorphic topological vector bundles if and only if they differ by a co-boundary. Thus, the topological vector bundles on T of rank r are “classified” by H1(T, GL(r, C)). -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Cox Rings and Partial Amplitude Permalink https://escholarship.org/uc/item/7bs989g2 Author Brown, Morgan Veljko Publication Date 2012 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Cox Rings and Partial Amplitude by Morgan Veljko Brown A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, BERKELEY Committee in charge: Professor David Eisenbud, Chair Professor Martin Olsson Professor Alistair Sinclair Spring 2012 Cox Rings and Partial Amplitude Copyright 2012 by Morgan Veljko Brown 1 Abstract Cox Rings and Partial Amplitude by Morgan Veljko Brown Doctor of Philosophy in Mathematics University of California, BERKELEY Professor David Eisenbud, Chair In algebraic geometry, we often study algebraic varieties by looking at their codimension one subvarieties, or divisors. In this thesis we explore the relationship between the global geometry of a variety X over C and the algebraic, geometric, and cohomological properties of divisors on X. Chapter 1 provides background for the results proved later in this thesis. There we give an introduction to divisors and their role in modern birational geometry, culminating in a brief overview of the minimal model program. In chapter 2 we explore criteria for Totaro's notion of q-amplitude. A line bundle L on X is q-ample if for every coherent sheaf F on X, there exists an integer m0 such that m ≥ m0 implies Hi(X; F ⊗ O(mL)) = 0 for i > q. -
Introduction to Algebraic Geometry
Introduction to Algebraic Geometry Jilong Tong December 6, 2012 2 Contents 1 Algebraic sets and morphisms 11 1.1 Affine algebraic sets . 11 1.1.1 Some definitions . 11 1.1.2 Hilbert's Nullstellensatz . 12 1.1.3 Zariski topology on an affine algebraic set . 14 1.1.4 Coordinate ring of an affine algebraic set . 16 1.2 Projective algebraic sets . 19 1.2.1 Definitions . 19 1.2.2 Homogeneous Nullstellensatz . 21 1.2.3 Homogeneous coordinate ring . 22 1.2.4 Exercise: plane curves . 22 1.3 Morphisms of algebraic sets . 24 1.3.1 Affine case . 24 1.3.2 Quasi-projective case . 26 2 The Language of schemes 29 2.1 Sheaves and locally ringed spaces . 29 2.1.1 Sheaves on a topological spaces . 29 2.1.2 Ringed space . 34 2.2 Schemes . 36 2.2.1 Definition of schemes . 36 2.2.2 Morphisms of schemes . 40 2.2.3 Projective schemes . 43 2.3 First properties of schemes and morphisms of schemes . 49 2.3.1 Topological properties . 49 2.3.2 Noetherian schemes . 50 2.3.3 Reduced and integral schemes . 51 2.3.4 Finiteness conditions . 53 2.4 Dimension . 54 2.4.1 Dimension of a topological space . 54 2.4.2 Dimension of schemes and rings . 55 2.4.3 The noetherian case . 57 2.4.4 Dimension of schemes over a field . 61 2.5 Fiber products and base change . 62 2.5.1 Sum of schemes . 62 2.5.2 Fiber products of schemes . -
Arxiv:1708.05877V4 [Math.AC]
NORMAL HYPERPLANE SECTIONS OF NORMAL SCHEMES IN MIXED CHARACTERISTIC JUN HORIUCHI AND KAZUMA SHIMOMOTO Abstract. The aim of this article is to prove that, under certain conditions, an affine flat normal scheme that is of finite type over a local Dedekind scheme in mixed characteristic admits infinitely many normal effective Cartier divisors. For the proof of this result, we prove the Bertini theorem for normal schemes of some type. We apply the main result to prove a result on the restriction map of divisor class groups of Grothendieck-Lefschetz type in mixed characteristic. Dedicated to Prof. Gennady Lyubeznik on the occasion of his 60th birthday. 1. Introduction Let X be a connected Noetherian normal scheme. Then does X have sufficiently many normal Cartier divisors? The existence of such a divisor when X is a normal projective variety over an algebraically closed field is already known. This fact follows from the classical Bertini theorem due to Seidenberg. In birational geometry, it is often necessary to compare the singularities of X with the singularities of a divisor D X, which is ⊂ known as adjunction (see [12] for this topic). In this article, we prove some results related to this problem in the mixed characteristic case. The main result is formulated as follows (see Corollary 5.4 and Remark 5.5). Theorem 1.1. Let X be a normal connected affine scheme such that there is a surjective flat morphism of finite type X Spec A, where A is an unramified discrete valuation ring → of mixed characteristic p > 0. Assume that dim X 2, the generic fiber of X Spec A ≥ → is geometrically connected and the residue field of A is infinite. -
Moving Codimension-One Subvarieties Over Finite Fields
Moving codimension-one subvarieties over finite fields Burt Totaro In topology, the normal bundle of a submanifold determines a neighborhood of the submanifold up to isomorphism. In particular, the normal bundle of a codimension-one submanifold is trivial if and only if the submanifold can be moved in a family of disjoint submanifolds. In algebraic geometry, however, there are higher-order obstructions to moving a given subvariety. In this paper, we develop an obstruction theory, in the spirit of homotopy theory, which gives some control over when a codimension-one subvariety moves in a family of disjoint subvarieties. Even if a subvariety does not move in a family, some positive multiple of it may. We find a pattern linking the infinitely many obstructions to moving higher and higher multiples of a given subvariety. As an application, we find the first examples of line bundles L on smooth projective varieties over finite fields which are nef (L has nonnegative degree on every curve) but not semi-ample (no positive power of L is spanned by its global sections). This answers questions by Keel and Mumford. Determining which line bundles are spanned by their global sections, or more generally are semi-ample, is a fundamental issue in algebraic geometry. If a line bundle L is semi-ample, then the powers of L determine a morphism from the given variety onto some projective variety. One of the main problems of the minimal model program, the abundance conjecture, predicts that a variety with nef canonical bundle should have semi-ample canonical bundle [15, Conjecture 3.12]. -
Arxiv:1807.03665V3 [Math.AG]
DEMAILLY’S NOTION OF ALGEBRAIC HYPERBOLICITY: GEOMETRICITY, BOUNDEDNESS, MODULI OF MAPS ARIYAN JAVANPEYKAR AND LJUDMILA KAMENOVA Abstract. Demailly’s conjecture, which is a consequence of the Green–Griffiths–Lang con- jecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly’s conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly’s definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a pro- jective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that Aut(X) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore “weaker” notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green–Griffiths–Lang conjecture on hyperbolic projective varieties. 1. Introduction The aim of this paper is to provide evidence for Demailly’s conjecture which says that a projective algebraically hyperbolic variety over C is Kobayashi hyperbolic. We first define the notion of an algebraically hyperbolic projective scheme over an alge- braically closed field k of characteristic zero which is not assumed to be C, and could be Q, for example. Then we provide indirect evidence for Demailly’s conjecture by showing that algebraically hyperbolic schemes share many common features with Kobayashi hyperbolic complex manifolds. -
Arxiv:1706.04845V2 [Math.AG] 26 Jan 2020 Usos Ewudas Iet Hn .Batfrifrigu Bu [B Comments
RELATIVE SEMI-AMPLENESS IN POSITIVE CHARACTERISTIC PAOLO CASCINI AND HIROMU TANAKA Abstract. Given an invertible sheaf on a fibre space between projective varieties of positive characteristic, we show that fibre- wise semi-ampleness implies relative semi-ampleness. The same statement fails in characteristic zero. Contents 1. Introduction 2 1.1. Description of the proof 3 2. Preliminary results 6 2.1. Notation and conventions 6 2.2. Basic results 8 2.3. Dimension formulas for universally catenary schemes 11 2.4. Relative semi-ampleness 13 2.5. Relative Keel’s theorem 18 2.6. Thickening process 20 2.7. Alteration theorem for quasi-excellent schemes 24 3. (Theorem C)n−1 implies (Theorem A)n 26 4. Numerically trivial case 30 4.1. The case where the total space is normal 30 4.2. Normalisation of the base 31 4.3. The vertical case 32 arXiv:1706.04845v2 [math.AG] 26 Jan 2020 4.4. (Theorem A)n implies (Theoerem B)n 37 4.5. Generalisation to algebraic spaces 43 5. (Theorem A)n and (Theorem B)n imply (Theorem C)n 44 6. Proofofthemaintheorems 49 7. Examples 50 2010 Mathematics Subject Classification. 14C20, 14G17. Key words and phrases. relative semi-ample, positive characteristic. The first author was funded by EPSRC. The second author was funded by EP- SRC and the Grant-in-Aid for Scientific Research (KAKENHI No. 18K13386). We would like to thank Y. Gongyo, Z. Patakfalvi and S. Takagi for many useful dis- cussions. We would also like to thank B. Bhatt for informing us about [BS17]. -
Notes on Grassmannians
NOTES ON GRASSMANNIANS ANDERS SKOVSTED BUCH This is class notes under construction. We have not attempted to account for the history of the results covered here. 1. Construction of Grassmannians 1.1. The set of points. Let k = k be an algebraically closed field, and let kn be the vector space of column vectors with n coordinates. Given a non-negative integer m ≤ n, the Grassmann variety Gr(m, n) is defined as a set by Gr(m, n)= {Σ ⊂ kn | Σ is a vector subspace with dim(Σ) = m} . Our first goal is to show that Gr(m, n) has a structure of algebraic variety. 1.2. Space with functions. Let FR(n, m) = {A ∈ Mat(n × m) | rank(A) = m} be the set of all n × m matrices of full rank, and let π : FR(n, m) → Gr(m, n) be the map defined by π(A) = Span(A), the column span of A. We define a topology on Gr(m, n) be declaring the a subset U ⊂ Gr(m, n) is open if and only if π−1(U) is open in FR(n, m), and further declare that a function f : U → k is regular if and only if f ◦ π is a regular function on π−1(U). This gives Gr(m, n) the structure of a space with functions. ex:morphism Exercise 1.1. (1) The map π : FR(n, m) → Gr(m, n) is a morphism of spaces with functions. (2) Let X be a space with functions and φ : Gr(m, n) → X a map. -
Utah/Fall 2020 3. Abstract Varieties. the Categories of Affine and Quasi
Algebraic Geometry I (Math 6130) Utah/Fall 2020 3. Abstract Varieties. The categories of affine and quasi-affine varieties over k are (by definition) full subcategories of the category of sheaved spaces (X; OX ) over k. We now enlarge the category just enough within the category of sheaved spaces to define a category of abstract varieties analogous to the categories of differentiable or analytic manifolds. Varieties are Noetherian and locally affine and separated (analogous to Hausdorff), the latter defined via products, which are also used to define the notion of a proper (analogous to compact or complete) variety. Definition 3.1. A topology on a set X is Noetherian if every descending chain: X ⊇ X1 ⊇ X2 ⊇ · · · of closed sets eventually stabilizes (or every ascending chain of open sets stabilizes). Remarks. (a) A Noetherian topological space X is quasi-compact, i.e. every open cover of X has a finite subcover, but a quasi-compact space need not be Noetherian. (b) The Zariski topology on a quasi-affine variety is Noetherian. (c) It is possible for a topological space X to fail to be Noetherian even if it has a cover by open sets, each of which is Noetherian with the induced topology. If the open cover is finite, however, then X is Noetherian. Definition 3.2. A Noetherian topological space X is reducible if X = X1 [ X2 for closed sets X1;X2 properly contained in X. Otherwise it is irreducible. Remarks. (a) As we saw in x1, every Noetherian topological space X is a finite union of irreducible closed subsets, accounting for the irreducible components of X. -
Using Grothendieck Groups to Define and Relate the Hilbert and Chern Polynomials Axiomatically
USING GROTHENDIECK GROUPS TO DEFINE AND RELATE THE HILBERT AND CHERN POLYNOMIALS AXIOMATICALLY TAKUMI MURAYAMA Abstract. The Hilbert polynomial can be defined axiomatically as a group homo- morphism on the Grothendieck group K(X) of a projective variety X, satisfying certain properties. The Chern polynomial can be similarly defined. We there- fore define these rather abstract notions to try and find a nice description of this relationship. Introduction Let X = Pr be a complex projective variety over an algebraically closed field k with coordinate ring S = k[x0; : : : ; xr]. Recall that the Hilbert function of a finitely generated graded S-module M is defined as func hM (t) := dimk Mt; and that we have the following Theorem (Hilbert). There is a polynomial hM (t) such that func hM (t) = hM (t) for t 0: We call hM (t) the Hilbert polynomial of M. The Hilbert polynomial satisfies a nice property: given an exact sequence 0 −! M 0 −! M −! M 00 −! 0 of graded S-modules, where each homomorphism is graded, i.e., it respects the grading on the modules M; M 0;M 00, then (1) hM (t) = hM 0 (t) + hM 00 (t); by the fact that the sequence above is exact at each degree t, and by the rank-nullity theorem. The Hilbert polynomial therefore said to be \additive on exact sequences"; this is an extremely useful fact that enables us to compute the Hilbert polynomial in many cases. For example, we computed the following for homework: Example (HW8, #3). Let f(w; x; y; z) and g(w; x; y; z) be relatively prime nonzero homogeneous polynomials of degree a and b in S := k[w; x; y; z].