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Stable Vector Bundle
In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable bundles were defined by David Mumford in and later built upon by David Gieseker, Fedor Bogomolov, Thomas Bridgeland and many others. Motivation One of the motivations for analyzing stable vector bundles is their nice behavior in families. In fact, Moduli spaces of stable vector bundles can be constructed using the Quot scheme in many cases, whereas the stack of vector bundles \mathbfGL_n is an Artin stack whose underlying set is a single point. Here's an example of a family of vector bundles which degenerate poorly. If we tensor the Euler sequence of \mathbb^1 by \mathcal(1) there is an exact sequence0 \to \mathcal(-1) \to \mathcal\oplus \mathcal \to \mathcal(1) \to 0which represents a non-zero element in v \in \text^1(\mathcal(1),\m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ample
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety ''X'' amounts to understanding the different ways of mapping ''X'' into projective space. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor. In more detail, a line bundle is called basepoint-free if it has enough sections to give a morphism to projective space. A line bundle is semi-ample if some positive power of it is basepoint-free; semi-ampleness is a kind of "nonnegativity". More strongly, a line bun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product Bundle
In differential geometry, the tensor product of vector bundles ''E'', ''F'' (over same space X) is a vector bundle, denoted by ''E'' ⊗ ''F'', whose fiber over a point x \in X is the tensor product of vector spaces ''E''''x'' ⊗ ''F''''x''.To construct a tensor-product bundle over a paracompact base, first note the construction is clear for trivial bundles. For the general case, if the base is compact, choose ''E'' such that ''E'' ⊕ ''E'' is trivial. Choose ''F'' in the same way. Then let ''E'' ⊗ ''F'' be the subbundle of (''E'' ⊕ ''E'') ⊗ (''F'' ⊕ ''F'') with the desired fibers. Finally, use the approximation argument to handle a non-compact base. See Hatcher for a general direct approach. Example: If ''O'' is a trivial line bundle, then ''E'' ⊗ ''O'' = ''E'' for any ''E''. Example: ''E'' ⊗ ''E'' ∗ is canonically isomorphic to the endomorphism bundle End(''E''), where ''E'' ∗ is the dual bundle of ''E''. Example: A line bundle ''L'' has tensor inverse: in fac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serre Twist
In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k-dimensional subspaces of V, given a point in the Grassmannian corresponding to a k-dimensional vector subspace W \subseteq V, the fiber over W is the subspace W itself. In the case of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles. Because of this, the tautological bundle is important in the study of characteristic classes. Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible sheaf) is :\mathcal_(-1), the dual of the hyperplane bundle or Serre's twisting sheaf \mathcal_(1). The hype ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi ( Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic \chi was classically defined for the surfaces of polyhedra, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion-free Module
In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is ''torsion free'' if its torsion submodule is reduced to its zero element. In integral domains the regular elements of the ring are its nonzero elements, so in this case a torsion-free module is one such that zero is the only element annihilated by some non-zero element of the ring. Some authors work only over integral domains and use this condition as the definition of a torsion-free module, but this does not work well over more general rings, for if the ring contains zero-divisors then the only module satisfying this condition is the zero module. Examples of torsion-free modules Over a commutative ring ''R'' with total quotient ring ''K'', a module ''M'' is torsion-free if and only if Tor1(''K''/''R'',''M'') vanishes. Therefore flat modules, and in particular free and projective modules, are t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane Section
In mathematics, a hyperplane section of a subset ''X'' of projective space P''n'' is the intersection of ''X'' with some hyperplane ''H''. In other words, we look at the subset ''X''''H'' of those elements ''x'' of ''X'' that satisfy the single linear condition ''L'' = 0 defining ''H'' as a linear subspace. Here ''L'' or ''H'' can range over the dual projective space of non-zero linear forms in the homogeneous coordinates, up to scalar multiplication. From a geometrical point of view, the most interesting case is when ''X'' is an algebraic subvariety; for more general cases, in mathematical analysis, some analogue of the Radon transform applies. In algebraic geometry, assuming therefore that ''X'' is ''V'', a subvariety not lying completely in any ''H'', the hyperplane sections are algebraic sets with irreducible components all of dimension dim(''V'') − 1. What more can be said is addressed by a collection of results known collectively as Bertini's theorem. The topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables with coefficients in ''k'', that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of \mathbb^n. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial. If ''X'' is a projective variety defined by a homogeneous prime ideal ''I'', then the quotient ring :k _0, \ldots, x_nI is called the homogeneous coordinate ring of ''X''. Basic invariants of ''X'' such as the degree and the dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Scheme
In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology. Definition First, let ''X'' be an affine scheme of finite type over a field ''k''. Equivalently, ''X'' has a closed immersion into affine space ''An'' over ''k'' for some natural number ''n''. Then ''X'' is the closed subscheme defined by some equations ''g''1 = 0, ..., ''g''''r'' = 0, where each ''gi'' is in the polynomial ring ''k'' 'x''1,..., ''x''''n'' The affine scheme ''X'' is smooth of dimension ''m'' over ''k'' if ''X'' has dimension at least ''m'' in a neighborhood of each point, and the matrix of derivatives (∂''g''''i''/∂''x''''j'') has rank at least ''n''−''m'' everywhere on ''X''. (It follows that ''X'' has dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Motivation Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space correspond to solutions of geometric problems. Here different solutions are identified if they a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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