The Concept of a Simple Point of an Abstract Algebraic Variety
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
[Math.CV] 6 May 2005
HOLOMORPHIC FLEXIBILITY PROPERTIES OF COMPLEX MANIFOLDS FRANC FORSTNERICˇ Abstract. We obtain results on approximation of holomorphic maps by al- gebraic maps, the jet transversality theorem for holomorphic and algebraic maps between certain classes of manifolds, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds. 1. Introduction In the present paper we use the term holomorphic flexibility property for any of several analytic properties of complex manifolds which are opposite to Koba- yashi-Eisenman-Brody hyperbolicity, the latter expressing holomorphic rigidity. A connected complex manifold Y is n-hyperbolic for some n ∈{1,..., dim Y } if every entire holomorphic map Cn → Y has rank less than n; for n = 1 this means that every holomorphic map C → Y is constant ([4], [16], [46], [47]). On the other hand, all flexibility properties of Y will require the existence of many such maps. We shall center our discussion around the following classical property which was studied by many authors (see the surveys [53] and [25]): Oka property: Every continuous map f0 : X → Y from a Stein manifold X is homotopic to a holomorphic map f1 : X → Y ; if f0 is holomorphic on (a neighbor- hood of) a compact H(X)-convex subset K of X then a homotopy {ft}t∈[0,1] from f0 to a holomorphic map f1 can be chosen such that ft is holomorphic and uniformly close to f0 on K for every t ∈ [0, 1]. Here, H(X)-convexity means convexity with respect to the algebra H(X) of all holomorphic functions on X (§2). -
Moduli Spaces
spaces. Moduli spaces such as the moduli of elliptic curves (which we discuss below) play a central role Moduli Spaces in a variety of areas that have no immediate link to the geometry being classified, in particular in alge- David D. Ben-Zvi braic number theory and algebraic topology. Moreover, the study of moduli spaces has benefited tremendously in recent years from interactions with Many of the most important problems in mathemat- physics (in particular with string theory). These inter- ics concern classification. One has a class of math- actions have led to a variety of new questions and new ematical objects and a notion of when two objects techniques. should count as equivalent. It may well be that two equivalent objects look superficially very different, so one wishes to describe them in such a way that equiv- 1 Warmup: The Moduli Space of Lines in alent objects have the same description and inequiva- the Plane lent objects have different descriptions. Moduli spaces can be thought of as geometric solu- Let us begin with a problem that looks rather simple, tions to geometric classification problems. In this arti- but that nevertheless illustrates many of the impor- cle we shall illustrate some of the key features of mod- tant ideas of moduli spaces. uli spaces, with an emphasis on the moduli spaces Problem. Describe the collection of all lines in the of Riemann surfaces. (Readers unfamiliar with Rie- real plane R2 that pass through the origin. mann surfaces may find it helpful to begin by reading about them in Part III.) In broad terms, a moduli To save writing, we are using the word “line” to mean problem consists of three ingredients. -
Arxiv:Math/0608115V2 [Math.NA] 13 Aug 2006 H Osrcino Nt Lmn Cee N Oo.I H Re the the in Is On
Multivariate Lagrange Interpolation and an Application of Cayley-Bacharach Theorem for It Xue-Zhang Lianga Jie-Lin Zhang Ming Zhang Li-Hong Cui Institute of Mathematics, Jilin University, Changchun, 130012, P. R. China [email protected] Abstract In this paper,we deeply research Lagrange interpolation of n-variables and give an application of Cayley-Bacharach theorem for it. We pose the concept of sufficient inter- section about s(1 ≤ s ≤ n) algebraic hypersurfaces in n-dimensional complex Euclidean space and discuss the Lagrange interpolation along the algebraic manifold of sufficient intersection. By means of some theorems ( such as Bezout theorem, Macaulay theorem (n) and so on ) we prove the dimension for the polynomial space Pm along the algebraic manifold S = s(f1, · · · ,fs)(where f1(X) = 0, · · · ,fs(X) = 0 denote s algebraic hyper- surfaces ) of sufficient intersection and give a convenient expression for dimension calcu- lation by using the backward difference operator. According to Mysovskikh theorem, we give a proof of the existence and a characterizing condition of properly posed set of nodes of arbitrary degree for interpolation along an algebraic manifold of sufficient intersec- tion. Further we point out that for s algebraic hypersurfaces f1(X) = 0, · · · ,fs(X) = 0 of sufficient intersection, the set of polynomials f1, · · · ,fs must constitute the H-base of ideal Is =<f1, · · · ,fs >. As a main result of this paper, we deduce a general method of constructing properly posed set of nodes for Lagrange interpolation along an algebraic manifold, namely the superposition interpolation process. At the end of the paper, we use the extended Cayley-Bacharach theorem to resolve some problems of Lagrange interpolation along the 0-dimensional and 1-dimensional algebraic manifold. -
Mathematisches Forschungsinstitut Oberwolfach Singularities
Mathematisches Forschungsinstitut Oberwolfach Report No. 43/2009 DOI: 10.4171/OWR/2009/43 Singularities Organised by Andr´as N´emethi, Budapest Duco van Straten, Mainz Victor A. Vassiliev, Moscow September 20th – September 26th, 2009 Abstract. Local/global Singularity Theory is concerned with the local/global structure of maps and spaces that occur in differential topology or theory of algebraic or analytic varieties. It uses methods from algebra, topology, alge- braic geometry and multi-variable complex analysis. Mathematics Subject Classification (2000): 14Bxx, 32Sxx, 58Kxx. Introduction by the Organisers The workshop Singularity Theory took place from September 20 to 26, 2009, and continued a long sequence of workshops Singularit¨aten that were organized regu- larly at Oberwolfach. It was attended by 46 participants with broad geographic representation. Funding from the Marie Curie Programme of EU provided com- plementary support for young researchers and PhD students. The schedule of the meeting comprised 23 lectures of one hour each, presenting recent progress and interesting directions in singularity theory. Some of the talks gave an overview of the state of the art, open problems and new efforts and results in certain areas of the field. For example, B. Teissier reported about the Kyoto meeting on ‘Resolution of Singularities’ and about recent developments in the geometry of local uniformization. J. Sch¨urmann presented the general picture of various generalizations of classical characteristic classes and the existence of functors connecting different geometrical levels. Strong applications of this for hypersurfaces was provided by L. Maxim. M. Kazarian reported on his new results and construction about the Thom polynomial of contact singularities; R. -
§4 Grassmannians 73
§4 Grassmannians 4.1 Plücker quadric and Gr(2,4). Let 푉 be 4-dimensional vector space. e set of all 2-di- mensional vector subspaces 푈 ⊂ 푉 is called the grassmannian Gr(2, 4) = Gr(2, 푉). More geomet- rically, the grassmannian Gr(2, 푉) is the set of all lines ℓ ⊂ ℙ = ℙ(푉). Sending 2-dimensional vector subspace 푈 ⊂ 푉 to 1-dimensional subspace 훬푈 ⊂ 훬푉 or, equivalently, sending a line (푎푏) ⊂ ℙ(푉) to 한 ⋅ 푎 ∧ 푏 ⊂ 훬푉 , we get the Plücker embedding 픲 ∶ Gr(2, 4) ↪ ℙ = ℙ(훬 푉). (4-1) Its image consists of all decomposable¹ grassmannian quadratic forms 휔 = 푎∧푏 , 푎, 푏 ∈ 푉. Clearly, any such a form has zero square: 휔 ∧ 휔 = 푎 ∧ 푏 ∧ 푎 ∧ 푏 = 0. Since an arbitrary form 휉 ∈ 훬푉 can be wrien¹ in appropriate basis of 푉 either as 푒 ∧ 푒 or as 푒 ∧ 푒 + 푒 ∧ 푒 and in the laer case 휉 is not decomposable, because of 휉 ∧ 휉 = 2 푒 ∧ 푒 ∧ 푒 ∧ 푒 ≠ 0, we conclude that 휔 ∈ 훬 푉 is decomposable if an only if 휔 ∧ 휔 = 0. us, the image of (4-1) is the Plücker quadric 푃 ≝ { 휔 ∈ 훬푉 | 휔 ∧ 휔 = 0 } (4-2) If we choose a basis 푒, 푒, 푒, 푒 ∈ 푉, the monomial basis 푒 = 푒 ∧ 푒 in 훬 푉, and write 푥 for the homogeneous coordinates along 푒, then straightforward computation 푥 ⋅ 푒 ∧ 푒 ∧ 푥 ⋅ 푒 ∧ 푒 = 2 푥 푥 − 푥 푥 + 푥 푥 ⋅ 푒 ∧ 푒 ∧ 푒 ∧ 푒 < < implies that 푃 is given by the non-degenerated quadratic equation 푥푥 = 푥푥 + 푥푥 . -
Arxiv:Math/0005288V1 [Math.QA] 31 May 2000 Ilb Setal H Rjciecodnt Igo H Variety
Mannheimer Manuskripte 254 math/0005288 SINGULAR PROJECTIVE VARIETIES AND QUANTIZATION MARTIN SCHLICHENMAIER Abstract. By the quantization condition compact quantizable K¨ahler mani- folds can be embedded into projective space. In this way they become projec- tive varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This allows for generalization to the case of singular vari- eties. The set-up is explained in the first part of the contribution. The second part of the contribution is of tutorial nature. Necessary notions, concepts, and results of algebraic geometry appearing in this approach to quantization are explained. In particular, the notions of projective varieties, embeddings, sin- gularities, and quotients appearing in geometric invariant theory are recalled. Contents Introduction 1 1. From quantizable compact K¨ahler manifolds to projective varieties 3 2. Projective varieties 6 2.1. The definition of a projective variety 6 2.2. Embeddings into Projective Space 9 2.3. The projective coordinate ring 12 3. Singularities 14 4. Quotients 17 4.1. Quotients in algebraic geometry 17 4.2. The relation with the symplectic quotient 19 References 20 Introduction arXiv:math/0005288v1 [math.QA] 31 May 2000 Compact K¨ahler manifolds which are quantizable, i.e. which admit a holomor- phic line bundle with curvature form equal to the K¨ahler form (a so called quantum line bundle) are projective algebraic manifolds. This means that with the help of the global holomorphic sections of a suitable tensor power of the quantum line bundle they can be embedded into a projective space of certain dimension. -
The Symplectic Topology of Projective Manifolds with Small Dual 3
THE SYMPLECTIC TOPOLOGY OF PROJECTIVE MANIFOLDS WITH SMALL DUAL PAUL BIRAN AND YOCHAY JERBY Abstract. We study smooth projective varieties with small dual variety using methods from symplectic topology. For such varieties we prove that the hyperplane class is an invertible element in the quantum cohomology of their hyperplane sections. We also prove that the affine part of such varieties are subcritical. We derive several topological and algebraic geometric consequences from that. The main tool in our work is the Seidel representation associated to Hamiltonian fibrations. 1. introduction and summary of the main results In this paper we study a special class of complex algebraic manifolds called projective manifolds with small dual. A projectively embedded algebraic manifold X ⊂ CP N is said to have small dual if the dual variety X∗ ⊂ (CP N )∗ has (complex) codimension ≥ 2. Recall that the dual variety X∗ of a projectively embedded algebraic manifold X ⊂ CP N is by definition the space of all hyperplanes H ⊂ CP N that are not transverse to X, i.e. X∗ = {H ∈ (CP N )∗ | H is somewhere tangent to X}. Let us mention that for “most” manifolds the codimension of X∗ is 1, however in special situations the codimension might be larger. To measure to which extent X deviates from the typical case one defines the defect 1 of an algebraic manifold X ⊂ CP N by ∗ def(X) = codimC(X ) − 1. arXiv:1107.0174v2 [math.AG] 27 Jun 2012 Thus we will call manifolds with small dual also manifolds with positive defect. Note that this is not an intrinsic property of X, but rather of a given projective embedding of X. -
Chapter 2 Affine Algebraic Geometry
Chapter 2 Affine Algebraic Geometry 2.1 The Algebraic-Geometric Dictionary The correspondence between algebra and geometry is closest in affine algebraic geom- etry, where the basic objects are solutions to systems of polynomial equations. For many applications, it suffices to work over the real R, or the complex numbers C. Since important applications such as coding theory or symbolic computation require finite fields, Fq , or the rational numbers, Q, we shall develop algebraic geometry over an arbitrary field, F, and keep in mind the important cases of R and C. For algebraically closed fields, there is an exact and easily motivated correspondence be- tween algebraic and geometric concepts. When the field is not algebraically closed, this correspondence weakens considerably. When that occurs, we will use the case of algebraically closed fields as our guide and base our definitions on algebra. Similarly, the strongest and most elegant results in algebraic geometry hold only for algebraically closed fields. We will invoke the hypothesis that F is algebraically closed to obtain these results, and then discuss what holds for arbitrary fields, par- ticularly the real numbers. Since many important varieties have structures which are independent of the field of definition, we feel this approach is justified—and it keeps our presentation elementary and motivated. Lastly, for the most part it will suffice to let F be R or C; not only are these the most important cases, but they are also the sources of our geometric intuitions. n Let A denote affine n-space over F. This is the set of all n-tuples (t1,...,tn) of elements of F. -
Math 632: Algebraic Geometry Ii Cohomology on Algebraic Varieties
MATH 632: ALGEBRAIC GEOMETRY II COHOMOLOGY ON ALGEBRAIC VARIETIES LECTURES BY PROF. MIRCEA MUSTA¸TA;˘ NOTES BY ALEKSANDER HORAWA These are notes from Math 632: Algebraic geometry II taught by Professor Mircea Musta¸t˘a in Winter 2018, LATEX'ed by Aleksander Horawa (who is the only person responsible for any mistakes that may be found in them). This version is from May 24, 2018. Check for the latest version of these notes at http://www-personal.umich.edu/~ahorawa/index.html If you find any typos or mistakes, please let me know at [email protected]. The problem sets, homeworks, and official notes can be found on the course website: http://www-personal.umich.edu/~mmustata/632-2018.html This course is a continuation of Math 631: Algebraic Geometry I. We will assume the material of that course and use the results without specific references. For notes from the classes (similar to these), see: http://www-personal.umich.edu/~ahorawa/math_631.pdf and for the official lecture notes, see: http://www-personal.umich.edu/~mmustata/ag-1213-2017.pdf The focus of the previous part of the course was on algebraic varieties and it will continue this course. Algebraic varieties are closer to geometric intuition than schemes and understanding them well should make learning schemes later easy. The focus will be placed on sheaves, technical tools such as cohomology, and their applications. Date: May 24, 2018. 1 2 MIRCEA MUSTA¸TA˘ Contents 1. Sheaves3 1.1. Quasicoherent and coherent sheaves on algebraic varieties3 1.2. Locally free sheaves8 1.3. -
Approximation of the Distance from a Point to an Algebraic Manifold
Approximation of the Distance from a Point to an Algebraic Manifold Alexei Yu. Uteshev and Marina V. Goncharova Faculty of Applied Mathematics, St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia Keywords: Algebraic Manifold, Distance Approximation, Discriminant, Level Set. 2 Abstract: The problem of geometric distance d evaluation from a point X0 to an algebraic curve in R or manifold 3 G(X) = 0 in R is treated in the form of comparison of exact value with two its successive approximations d(1) and d(2). The geometric distance is evaluated from the univariate distance equation possessing the zero 2 set coinciding with that of critical values of the function d (X0), while d(1)(X0) and d(2)(X0) are obtained 2 via expansion of d (X0) into the power series of the algebraic distance G(X0). We estimate the quality of approximation comparing the relative positions of the level sets of d(X), d(1)(X) and d(2)(X). 1 INTRODUCTION Approximation (2) is known as Sampson’s dis- tance (Sampson, 1982). We aim at comparison of We treat the problem of Euclidean distance evaluation the qualities of approximations (2) and (3). We deal from a point X0 to manifold defined implicitly by the mostly with the case of algebraic manifolds (1), i.e. equation G(X) 2 R[X]. For this case, the tolerances of the ap- G(X) = 0 (1) proximations can be evaluated via comparison with n the true (geometric) distance value determined from in R ;n 2 f2;3g. Here G(X) is twice differentiable real valued function and it is assumed that the equa- the so-called distance equation (Uteshev and Gon- n charova, 2017; Uteshev and Goncharova, 2018). -
Linear Algebraic Groups
Clay Mathematics Proceedings Volume 4, 2005 Linear Algebraic Groups Fiona Murnaghan Abstract. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups. 1. Algebraic groups Let K be an algebraically closed field. An algebraic K-group G is an algebraic variety over K, and a group, such that the maps µ : G × G → G, µ(x, y) = xy, and ι : G → G, ι(x)= x−1, are morphisms of algebraic varieties. For convenience, in these notes, we will fix K and refer to an algebraic K-group as an algebraic group. If the variety G is affine, that is, G is an algebraic set (a Zariski-closed set) in Kn for some natural number n, we say that G is a linear algebraic group. If G and G′ are algebraic groups, a map ϕ : G → G′ is a homomorphism of algebraic groups if ϕ is a morphism of varieties and a group homomorphism. Similarly, ϕ is an isomorphism of algebraic groups if ϕ is an isomorphism of varieties and a group isomorphism. A closed subgroup of an algebraic group is an algebraic group. If H is a closed subgroup of a linear algebraic group G, then G/H can be made into a quasi- projective variety (a variety which is a locally closed subset of some projective space). If H is normal in G, then G/H (with the usual group structure) is a linear algebraic group. Let ϕ : G → G′ be a homomorphism of algebraic groups. Then the kernel of ϕ is a closed subgroup of G and the image of ϕ is a closed subgroup of G. -
Complex Algebraic Geometry
Complex Algebraic Geometry Jean Gallier∗ and Stephen S. Shatz∗∗ ∗Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] ∗∗Department of Mathematics University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] February 25, 2011 2 Contents 1 Complex Algebraic Varieties; Elementary Theory 7 1.1 What is Geometry & What is Complex Algebraic Geometry? . .......... 7 1.2 LocalStructureofComplexVarieties. ............ 14 1.3 LocalStructureofComplexVarieties,II . ............. 28 1.4 Elementary Global Theory of Varieties . ........... 42 2 Cohomologyof(Mostly)ConstantSheavesandHodgeTheory 73 2.1 RealandComplex .................................... ...... 73 2.2 Cohomology,deRham,Dolbeault. ......... 78 2.3 Hodge I, Analytic Preliminaries . ........ 89 2.4 Hodge II, Globalization & Proof of Hodge’s Theorem . ............ 107 2.5 HodgeIII,TheK¨ahlerCase . .......... 131 2.6 Hodge IV: Lefschetz Decomposition & the Hard Lefschetz Theorem............... 147 2.7 ExtensionsofResultstoVectorBundles . ............ 162 3 The Hirzebruch-Riemann-Roch Theorem 165 3.1 Line Bundles, Vector Bundles, Divisors . ........... 165 3.2 ChernClassesandSegreClasses . .......... 179 3.3 The L-GenusandtheToddGenus .............................. 215 3.4 CobordismandtheSignatureTheorem. ........... 227 3.5 The Hirzebruch–Riemann–Roch Theorem (HRR) . ............ 232 3 4 CONTENTS Preface This manuscript is based on lectures given by Steve Shatz for the course Math 622/623–Complex Algebraic Geometry, during Fall 2003 and Spring 2004. The process for producing this manuscript was the following: I (Jean Gallier) took notes and transcribed them in LATEX at the end of every week. A week later or so, Steve reviewed these notes and made changes and corrections. After the course was over, Steve wrote up additional material that I transcribed into LATEX. The following manuscript is thus unfinished and should be considered as work in progress.