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Vector Bundles and Projective Varieties

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Vector Bundles and Projective Varieties VECTOR BUNDLES AND PROJECTIVE VARIETIES by NICHOLAS MARINO Submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics, Applied Mathematics, and Statistics CASE WESTERN RESERVE UNIVERSITY January 2019 CASE WESTERN RESERVE UNIVERSITY Department of Mathematics, Applied Mathematics, and Statistics We hereby approve the thesis of Nicholas Marino Candidate for the degree of Master of Science Committee Chair Nick Gurski Committee Member David Singer Committee Member Joel Langer Date of Defense: 10 December, 2018 1 Contents Abstract 3 1 Introduction 4 2 Basic Constructions 5 2.1 Elementary Definitions . 5 2.2 Line Bundles . 8 2.3 Divisors . 12 2.4 Differentials . 13 2.5 Chern Classes . 14 3 Moduli Spaces 17 3.1 Some Classifications . 17 3.2 Stable and Semi-stable Sheaves . 19 3.3 Representability . 21 4 Vector Bundles on Pn 26 4.1 Cohomological Tools . 26 4.2 Splitting on Higher Projective Spaces . 27 4.3 Stability . 36 5 Low-Dimensional Results 37 5.1 2-bundles and Surfaces . 37 5.2 Serre's Construction and Hartshorne's Conjecture . 39 5.3 The Horrocks-Mumford Bundle . 42 6 Ulrich Bundles 44 7 Conclusion 48 8 References 50 2 Vector Bundles and Projective Varieties Abstract by NICHOLAS MARINO Vector bundles play a prominent role in the study of projective algebraic varieties. Vector bundles can describe facets of the intrinsic geometry of a variety, as well as its relationship to other varieties, especially projective spaces. Here we outline the general theory of vector bundles and describe their classification and structure. We also consider some special bundles and general results in low dimensions, especially rank 2 bundles and surfaces, as well as bundles on projective spaces. Finally, we indicate some open problems and current areas of research. 3 1 Introduction There are several aspects by which vector bundles are of use in algebraic geome- try. Historically, the most straightfoward is the geometric one: vector bundles are higher-dimensional varieties whose local structure is only truly interesting in a lower dimension. In this way, they provide digestible examples of varieties in higher dimen- sion. On the other hand, from this point of view vector bundles are slightly artificial in algebraic geometry: other classes of varieties with fibered structures aren't so strin- gent in the restrictions they put on their fibrations, e.g. elliptic surfaces, which might have singular fibers. There are also, of course, algebraic parts of the theory. As the sheaves associated to free modules of finite rank, they allow for the use of linear al- gebra in a milieu of considerable sophistication (the smug scheme theorist might say that the study of vector bundles began with the case where X = Spec k, for k any field). On the other hand, and perhaps more importantly, there are topological consider- ations. Given the development of abstract schemes and varieties obtained by gluing affine ones together in a particular manner, it is then unsurprising to find a sort of fiber bundle construction on them which glues together in the simplest possible way. From this point of view it is almost obvious, even inevitable, that the study of algebraic vector bundles would begin in ernest in that radical period of abstrac- tion between the ideas of Weil and Grothendieck. This study would be bolstered by the introduction of sheaf-theoretic methods into algebraic geometry. Indeed, though Serre's seminal paper \Faisceaux Alg´ebriqueCoher´ents" [Ser55] was principally con- cerened with coherent sheaves, the locally free sheaves make up a prominent class of concrete coherent sheaves which are easy to describe, manipulate, and work with. Vector bundles also pose categorical problems for mathematicians. For instance, 4 the construction of moduli spaces of various classes of objects is of central concern to some, and the study of moduli spaces of vector bundles provides both a fertile testbed for this theory, as well as several open problems. Along the same line, understanding how the underlying geometry of a projective variety affects the range of possible vector bundles on it has occupied the work of many mathematicians over the last epoch, and continues to do so today. 2 Basic Constructions 2.1 Elementary Definitions In what follows X is a projective scheme or variety, which is to be taken as meaning a complete, separated, Noetherian, integral scheme of finite type over an algebraically closed field k of characteristic 0. Sometimes X is not assumed to be integral, and may not be either reduced or irreducible, but in most of those cass will be in the context of X being a closed subscheme of another scheme, and the need for this generality will be apparent. Definition. A linear fibration of rank r over X is a projective variety E together with a surjective morphism π : E ! X such that the fibers π−1(x) have the structure of a vector space of dimension r for all x 2 X. The trivial fibration is the product π : X ×Ar ! X, π the projection map. A vector bundle over X is a linear fibration π : E ! X which is locally trivial, i.e., there exists an open cover fUig of X, and −1 ∼ n −1 r r isomorphisms 'i : π (Ui) −! Ui ×A such that 'i ◦'j : Ui \Uj ×A ! Ui \Uj ×A is a map of the form (x; v) 7! (x; g;i;j(x)(v)), where gi;j : Ui \ Uj ! GL(r; k) is a morphism of varieties. The gi;j are the transition functions of the bundle. These also satisfy the conditions that gi;i = IdAr , and on the intersection Ui \ Uj \ Uk, 5 gi;k = gi;jgj;k. [Pot97] A morphism f : E ! F of vector bundles π : E ! X and π0 : F ! X is a 0 −1 morphism of varieties such that π ◦ f = π, and the maps on the fibers fx : π (x) ! π0−1(x) is a linear map. One could alternatively (and often more conveniently) treat vector bundles as follows: Definition. Let X be a projective variety with structure sheaf OX .A vector bundle of rank r over X is a locally free OX -module E of rank r; i.e., an OX -module E and an open cover fUig of X such that E(Ui) is a free OX (U)-module of rank r. Given a vector bundle π : E ! X, the sections Γ(U; E) of E over U ⊆ X (i.e., regular maps s : U ! E with π ◦ s = IdU ) form an OX (U)-module, given by (s + t)(x) = s(x) + t(x) and (α · s)(x) = α(x)s(x) for s; t 2 Γ(U; E) and α 2 OX (U). On trivializing open sets Ui, this module is free, so one can obtain a locally free sheaf of rank r by this method, often denoted O(E). On the other hand, given such a locally free sheaf E , for each x 2 X let mx denote the maximal ideal of the local ring Ox;X , and let Ex denote the Ox;X -module given by the stalk of E at x. Then Ex=mxEx ` is a (Ox;X =mx)-vector space, and E = x2X Ex=mxEx ! X can be given the structure of a vector bundle on X such that O(E) ∼= E , and in fact these constructions give an equivalence of categories: the category of vector bundles π : E ! X over X and morphisms of vector bundles with the category of locally free OX -modules and morphisms of OX -modules. However, there are several natural constructions on vector spaces that admit gen- eralizations to vector bundles, which are easier to define for locally free sheaves than for locally trivial linear fibrations. If one wishes to give these definitions for such bun- dles π : E ! X, this mostly involves performing the construction on the pointwise 6 fibers Ex and then showing these can be sewn together into a locally trivial bundle. To avoid this, one can simply consider the vector bundles as coherent sheaves, and use the extant definitions there. In light of this, one may define the direct sum (also called the Whitney sum) of vector bundles E and F , denoted E ⊕ F , to be their direct sum as OX -modules. Similarly one can define the tensor product E ⊗ F , the dual bundle E_, the exterior powers ^kE, and the symmetric powers SymkE.A sub-bundle F of a locally free OX -module E is a locally free sub-module; if E ⊆ F is a sub-bundle, the quotient bundle E=F is well-defined. In the case of vector bundles π : E ! X and π0 : F ! X, and a map of vector bundles f : E ! F , this map has well-defined kernel and image bundles exactly when the linear map fx is has constant rank over all x 2 X. Equivalently, this is the statement that a map of locally free OX -modules has a locally free kernel and cokernel exactly when the rank of this map is constant on the stalks. In general, this need not be the case, but if E and F are locally-free OX -modules, the kernel and cokernel of f are well-defined coherent sheaves, i.e., OX -modules F such that X admits a covering by open affine subsets fUig, on which F jUi is of the form ~ Mi for a finitely generated Γ(U; OX )-module Mi (One might also want to consider the category of quasi-coherent sheaves, which is the category of F as above, where the Mi are not assumed to be finitely generated [Ser55].
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