Algebraic Geometry
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Rational Curves on Hypersurfaces
Rational curves on hypersurfaces ||||||{ Lecture notes for the II Latin American School of Algebraic Geometry and Applications 1{12 June 2015 Cabo Frio, Brazil ||||||{ Olivier Debarre January 4, 2016 Contents 1 Projective spaces and Grassmannians 3 1.1 Projective spaces . 3 1.2 The Euler sequence . 4 1.3 Grassmannians . 4 1.4 Linear spaces contained in a subscheme of P(V )................ 5 2 Projective lines contained in a hypersurface 9 2.1 Local study of F (X) ............................... 9 2.2 Schubert calculus . 10 2.3 Projective lines contained in a hypersurface . 12 2.3.1 Existence of lines in a hypersurface . 12 2.3.2 Lines through a point . 16 2.3.3 Free lines . 17 2.4 Projective lines contained in a quadric hypersurface . 18 2.5 Projective lines contained in a cubic hypersurface . 19 2.5.1 Lines on a smooth cubic surface . 20 2.5.2 Lines on a smooth cubic fourfold . 22 2.6 Cubic hypersurfaces over finite fields . 25 3 Conics and curves of higher degrees 29 3.1 Conics in the projective space . 29 3.2 Conics contained in a hypersurface . 30 3.2.1 Conics contained in a quadric hypersurface . 32 3.2.2 Conics contained in a cubic hypersurface . 33 3.3 Rational curves of higher degrees . 35 4 Varieties covered by rational curves 39 4.1 Uniruled and separably uniruled varieties . 39 4.2 Free rational curves and separably uniruled varieties . 41 4.3 Minimal free rational curves . 44 Abstract In the past few decades, it has become more and more obvious that rational curves are the primary player in the study of the birational geometry of projective varieties. -
Divisors and the Picard Group
18.725 Algebraic Geometry I Lecture 15 Lecture 15: Divisors and the Picard Group Suppose X is irreducible. The (Weil) divisor DivW (X) is defined as the formal Z combinations of subvarieties of codimension 1. On the other hand, the Cartier divisor group, DivC (X), consists of subvariety locally given by a nonzero rational function defined up to multiplication by a nonvanishing function. Definition 1. An element of DivC (X) is given by 1. a covering Ui; and 2. Rational functions fi on Ui, fi 6= 0, ∗ such that on Ui \ Uj, fj = 'ijfi, where 'ij 2 O (Ui \ Uj). ∗ ∗ ∗ Another way to express this is that DivC (X) = Γ(K =O ), where K is the sheaf of nonzero rational functions, where O∗ is the sheaf of regular functions. Remark 1. Cartier divisors and invertible sheaves are equivalent (categorically). Given D 2 DivC (X), then we get an invertible subsheaf in K, locally it's fiO, the O-submodule generated by fi by construction it is locally isomorphic to O. Conversely if L ⊆ K is locally isomorphic to O, A system of local generators defines the data as above. Note that the abelian group structure on Γ(K∗=O∗) corresponds to multiplying by the ideals. ∗ ∗ ∗ ∗ Proposition 1. Pic(X) = DivC (X)=Im(K ) = Γ(K =O )= im Γ(K ). Proof. We already have a function DivC (X) = IFI ! Pic (IFI: invertible frational ideals) given by (L ⊆ K) 7! L. This map is an homomorphism. It is also onto: choosing a trivialization LjU = OjU gives an isomorphism ∼ ∗ ∗ L ⊗O⊇L K = K.No w let's look at its kernel: it consits of sections of K =O coming from O ⊆ K, which is just the same as the set of nonzero rational functions, which is im Γ(K∗) = Γ(K∗)=Γ(O∗). -
The Jouanolou-Thomason Homotopy Lemma
The Jouanolou-Thomason homotopy lemma Aravind Asok February 9, 2009 1 Introduction The goal of this note is to prove what is now known as the Jouanolou-Thomason homotopy lemma or simply \Jouanolou's trick." Our main reason for discussing this here is that i) most statements (that I have seen) assume unncessary quasi-projectivity hypotheses, and ii) most applications of the result that I know (e.g., in homotopy K-theory) appeal to the result as merely a \black box," while the proof indicates that the construction is quite geometric and relatively explicit. For simplicity, throughout the word scheme means separated Noetherian scheme. Theorem 1.1 (Jouanolou-Thomason homotopy lemma). Given a smooth scheme X over a regular Noetherian base ring k, there exists a pair (X;~ π), where X~ is an affine scheme, smooth over k, and π : X~ ! X is a Zariski locally trivial smooth morphism with fibers isomorphic to affine spaces. 1 Remark 1.2. In terms of an A -homotopy category of smooth schemes over k (e.g., H(k) or H´et(k); see [MV99, x3]), the map π is an A1-weak equivalence (use [MV99, x3 Example 2.4]. Thus, up to A1-weak equivalence, any smooth k-scheme is an affine scheme smooth over k. 2 An explicit algebraic form Let An denote affine space over Spec Z. Let An n 0 denote the scheme quasi-affine and smooth over 2m Spec Z obtained by removing the fiber over 0. Let Q2m−1 denote the closed subscheme of A (with coordinates x1; : : : ; x2m) defined by the equation X xixm+i = 1: i Consider the following simple situation. -
A Note on Presentation of General Linear Groups Over a Finite Field
Southeast Asian Bulletin of Mathematics (2019) 43: 217–224 Southeast Asian Bulletin of Mathematics c SEAMS. 2019 A Note on Presentation of General Linear Groups over a Finite Field Swati Maheshwari and R. K. Sharma Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India Email: [email protected]; [email protected] Received 22 September 2016 Accepted 20 June 2018 Communicated by J.M.P. Balmaceda AMS Mathematics Subject Classification(2000): 20F05, 16U60, 20H25 Abstract. In this article we have given Lie regular generators of linear group GL(2, Fq), n where Fq is a finite field with q = p elements. Using these generators we have obtained presentations of the linear groups GL(2, F2n ) and GL(2, Fpn ) for each positive integer n. Keywords: Lie regular units; General linear group; Presentation of a group; Finite field. 1. Introduction Suppose F is a finite field and GL(n, F) is the general linear the group of n × n invertible matrices and SL(n, F) is special linear group of n × n matrices with determinant 1. We know that GL(n, F) can be written as a semidirect product, GL(n, F)= SL(n, F) oF∗, where F∗ denotes the multiplicative group of F. Let H and K be two groups having presentations H = hX | Ri and K = hY | Si, then a presentation of semidirect product of H and K is given by, −1 H oη K = hX, Y | R,S,xyx = η(y)(x) ∀x ∈ X,y ∈ Y i, where η : K → Aut(H) is a group homomorphism. Now we summarize some literature survey related to the presentation of groups. -
Perverse Sheaves
Perverse Sheaves Bhargav Bhatt Fall 2015 1 September 8, 2015 The goal of this class is to introduce perverse sheaves, and how to work with it; plus some applications. Background For more background, see Kleiman's paper entitled \The development/history of intersection homology theory". On manifolds, the idea is that you can intersect cycles via Poincar´eduality|we want to be able to do this on singular spces, not just manifolds. Deligne figured out how to compute intersection homology via sheaf cohomology, and does not use anything about cycles|only pullbacks and truncations of complexes of sheaves. In any derived category you can do this|even in characteristic p. The basic summary is that we define an abelian subcategory that lives inside the derived category of constructible sheaves, which we call the category of perverse sheaves. We want to get to what is called the decomposition theorem. Outline of Course 1. Derived categories, t-structures 2. Six Functors 3. Perverse sheaves—definition, some properties 4. Statement of decomposition theorem|\yoga of weights" 5. Application 1: Beilinson, et al., \there are enough perverse sheaves", they generate the derived category of constructible sheaves 6. Application 2: Radon transforms. Use to understand monodromy of hyperplane sections. 7. Some geometric ideas to prove the decomposition theorem. If you want to understand everything in the course you need a lot of background. We will assume Hartshorne- level algebraic geometry. We also need constructible sheaves|look at Sheaves in Topology. Problem sets will be given, but not collected; will be on the webpage. There are more references than BBD; they will be online. -
Boundary and Shape of Cohen-Macaulay Cone
BOUNDARY AND SHAPE OF COHEN-MACAULAY CONE HAILONG DAO AND KAZUHIKO KURANO Abstract. Let R be a Cohen-Macaulay local domain. In this paper we study the cone of Cohen-Macaulay modules inside the Grothendieck group of finitely generated R-modules modulo numerical equivalences, introduced in [3]. We prove a result about the boundary of this cone for Cohen-Macaulay domain admitting de Jong’s alterations, and use it to derive some corollaries on finite- ness of isomorphism classes of maximal Cohen-Macaulay ideals. Finally, we explicitly compute the Cohen-Macaulay cone for certain isolated hypersurface singularities defined by ξη − f(x1; : : : ; xn). 1. Introduction Let R be a Noetherian local ring and G0(R) the Grothendieck group of finitely generated R-modules. Using Euler characteristic of perfect complexes with finite length homologies (a generalized version of Serre’s intersection multiplicity pair- ings), one could define the notion of numerical equivalence on G0(R) as in [17]. See Section 2 for precise definitions. When R is the local ring at the vertex of an affine cone over a smooth projective variety X, this notion can be deeply related to that of numerical equivalences on the Chow group of X as in [17] and [20]. Let G0(R) be the Grothendieck group of R modulo numerical equivalences. A simple result in homological algebra tells us that if M is maximal Cohen- Macaulay (MCM), the Euler characteristic function will always be positive. Thus, maximal Cohen-Macaulay modules all survive in G0(R), and it makes sense to talk about the cone of Cohen-Macaulay modules inside G (R) : 0 R X C (R) = [M] ⊂ G (R) : CM R≥0 0 R M:MCM The definition of this cone and some of its basic properties was given in [3]. -
Abelian Varieties with Complex Multiplication and Modular Functions, by Goro Shimura, Princeton Univ
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 3, Pages 405{408 S 0273-0979(99)00784-3 Article electronically published on April 27, 1999 Abelian varieties with complex multiplication and modular functions, by Goro Shimura, Princeton Univ. Press, Princeton, NJ, 1998, xiv + 217 pp., $55.00, ISBN 0-691-01656-9 The subject that might be called “explicit class field theory” begins with Kro- necker’s Theorem: every abelian extension of the field of rational numbers Q is a subfield of a cyclotomic field Q(ζn), where ζn is a primitive nth root of 1. In other words, we get all abelian extensions of Q by adjoining all “special values” of e(x)=exp(2πix), i.e., with x Q. Hilbert’s twelfth problem, also called Kronecker’s Jugendtraum, is to do something2 similar for any number field K, i.e., to generate all abelian extensions of K by adjoining special values of suitable special functions. Nowadays we would add that the reciprocity law describing the Galois group of an abelian extension L/K in terms of ideals of K should also be given explicitly. After K = Q, the next case is that of an imaginary quadratic number field K, with the real torus R/Z replaced by an elliptic curve E with complex multiplication. (Kronecker knew what the result should be, although complete proofs were given only later, by Weber and Takagi.) For simplicity, let be the ring of integers in O K, and let A be an -ideal. Regarding A as a lattice in C, we get an elliptic curve O E = C/A with End(E)= ;Ehas complex multiplication, or CM,by .If j=j(A)isthej-invariant ofOE,thenK(j) is the Hilbert class field of K, i.e.,O the maximal abelian unramified extension of K. -
Rigid Analytic Curves and Their Jacobians
Rigid analytic curves and their Jacobians Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakult¨at f¨urMathematik und Wirtschaftswissenschaften der Universit¨atUlm vorgelegt von Sophie Schmieg aus Ebersberg Ulm 2013 Erstgutachter: Prof. Dr. Werner Lutkebohmert¨ Zweitgutachter: Prof. Dr. Stefan Wewers Amtierender Dekan: Prof. Dr. Dieter Rautenbach Tag der Promotion: 19. Juni 2013 Contents Glossary of Notations vii Introduction ix 1. The Jacobian of a curve in the complex case . ix 2. Mumford curves and general rigid analytic curves . ix 3. Outline of the chapters and the results of this work . x 4. Acknowledgements . xi 1. Some background on rigid geometry 1 1.1. Non-Archimedean analysis . 1 1.2. Affinoid varieties . 2 1.3. Admissible coverings and rigid analytic varieties . 3 1.4. The reduction of a rigid analytic variety . 4 1.5. Adic topology and complete rings . 5 1.6. Formal schemes . 9 1.7. Analytification of an algebraic variety . 11 1.8. Proper morphisms . 12 1.9. Etale´ morphisms . 13 1.10. Meromorphic functions . 14 1.11. Examples . 15 2. The structure of a formal analytic curve 17 2.1. Basic definitions . 17 2.2. The formal fiber of a point . 17 2.3. The formal fiber of regular points and double points . 22 2.4. The formal fiber of a general singular point . 23 2.5. Formal blow-ups . 27 2.6. The stable reduction theorem . 29 2.7. Examples . 31 3. Group objects and Jacobians 33 3.1. Some definitions from category theory . 33 3.2. Group objects . 35 3.3. Central extensions of group objects . -
Foundations of Algebraic Geometry Class 13
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 13 RAVI VAKIL CONTENTS 1. Some types of morphisms: quasicompact and quasiseparated; open immersion; affine, finite, closed immersion; locally closed immersion 1 2. Constructions related to “smallest closed subschemes”: scheme-theoretic image, scheme-theoretic closure, induced reduced subscheme, and the reduction of a scheme 12 3. More finiteness conditions on morphisms: (locally) of finite type, quasifinite, (locally) of finite presentation 16 We now define a bunch of types of morphisms. (These notes include some topics dis- cussed the previous class.) 1. SOME TYPES OF MORPHISMS: QUASICOMPACT AND QUASISEPARATED; OPEN IMMERSION; AFFINE, FINITE, CLOSED IMMERSION; LOCALLY CLOSED IMMERSION In this section, we'll give some analogues of open subsets, closed subsets, and locally closed subsets. This will also give us an excuse to define affine and finite morphisms (closed immersions are a special case). It will also give us an excuse to define some im- portant special closed immersions, in the next section. In section after that, we'll define some more types of morphisms. 1.1. Quasicompact and quasiseparated morphisms. A morphism f : X Y is quasicompact if for every open affine subset U of Y, f-1(U) is quasicompact. Equivalently! , the preimage of any quasicompact open subset is quasicom- pact. We will like this notion because (i) we know how to take the maximum of a finite set of numbers, and (ii) most reasonable schemes will be quasicompact. 1.A. EASY EXERCISE. Show that the composition of two quasicompact morphisms is quasicompact. 1.B. EXERCISE. Show that any morphism from a Noetherian scheme is quasicompact. -
Algebraic Cycles and Fibrations
Documenta Math. 1521 Algebraic Cycles and Fibrations Charles Vial Received: March 23, 2012 Revised: July 19, 2013 Communicated by Alexander Merkurjev Abstract. Let f : X → B be a projective surjective morphism between quasi-projective varieties. The goal of this paper is the study of the Chow groups of X in terms of the Chow groups of B and of the fibres of f. One of the applications concerns quadric bundles. When X and B are smooth projective and when f is a flat quadric fibration, we show that the Chow motive of X is “built” from the motives of varieties of dimension less than the dimension of B. 2010 Mathematics Subject Classification: 14C15, 14C25, 14C05, 14D99 Keywords and Phrases: Algebraic cycles, Chow groups, quadric bun- dles, motives, Chow–K¨unneth decomposition. Contents 1. Surjective morphisms and motives 1526 2. OntheChowgroupsofthefibres 1530 3. A generalisation of the projective bundle formula 1532 4. On the motive of quadric bundles 1534 5. Onthemotiveofasmoothblow-up 1536 6. Chow groups of varieties fibred by varieties with small Chow groups 1540 7. Applications 1547 References 1551 Documenta Mathematica 18 (2013) 1521–1553 1522 Charles Vial For a scheme X over a field k, CHi(X) denotes the rational Chow group of i- dimensional cycles on X modulo rational equivalence. Throughout, f : X → B will be a projective surjective morphism defined over k from a quasi-projective variety X of dimension dX to an irreducible quasi-projective variety B of di- mension dB, with various extra assumptions which will be explicitly stated. Let h be the class of a hyperplane section in the Picard group of X. -
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. -
Brauer Groups of Abelian Schemes
ANNALES SCIENTIFIQUES DE L’É.N.S. RAYMOND T. HOOBLER Brauer groups of abelian schemes Annales scientifiques de l’É.N.S. 4e série, tome 5, no 1 (1972), p. 45-70 <http://www.numdam.org/item?id=ASENS_1972_4_5_1_45_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1972, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. scienL EC. Norm. Sup., 4® serie, t. 5, 1972, p. 45 ^ 70. BRAUER GROUPS OF ABELIAN SCHEMES BY RAYMOND T. HOOBLER 0 Let A be an abelian variety over a field /c. Mumford has given a very beautiful construction of the dual abelian variety in the spirit of Grothen- dieck style algebraic geometry by using the theorem of the square, its corollaries, and cohomology theory. Since the /c-points of Pic^n is H1 (A, G^), it is natural to ask how much of this work carries over to higher cohomology groups where the computations must be made in the etale topology to render them non-trivial. Since H2 (A, Gm) is essentially a torsion group, the representability of the corresponding functor does not have as much geometric interest as for H1 (A, G^).