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Degeneration (algebraic Geometry)
In algebraic geometry, a degeneration (or specialization) is the act of taking a limit of a family of varieties. Precisely, given a morphism :\pi: \mathcal \to C, of a variety (or a scheme) to a curve ''C'' with origin 0 (e.g., affine or projective line), the fibers :\pi^(t) form a family of varieties over ''C''. Then the fiber \pi^(0) may be thought of as the limit of \pi^(t) as t \to 0. One then says the family \pi^(t), t \ne 0 ''degenerates'' to the ''special'' fiber \pi^(0). The limiting process behaves nicely when \pi is a flat morphism and, in that case, the degeneration is called a flat degeneration. Many authors assume degenerations to be flat. When the family \pi^(t) is trivial away from a special fiber; i.e., \pi^(t) is independent of t \ne 0 up to (coherent) isomorphisms, \pi^(t), t \ne 0 is called a general fiber. Degenerations of curves In the study of moduli of curves, the important point is to understand the boundaries of the moduli, which amounts to understand deg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e., :f_P\colon \mathcal_ \to \mathcal_ is a flat map for all ''P'' in ''X''. A map of rings A\to B is called flat if it is a homomorphism that makes ''B'' a flat ''A''-module. A morphism of schemes is called faithfully flat if it is both surjective and flat. Two basic intuitions regarding flat morphisms are: *flatness is a generic property; and *the failure of flatness occurs on the jumping set of the morphism. The first of these comes from commutative algebra: subject to some finiteness conditions on ''f'', it can be shown that there is a non-empty open subscheme Y' of ''Y'', such that ''f'' restricted to ''Y''′ is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of the fiber product of schemes, applied to ''f'' and the inclusio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Of Curves
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme (mathematics), scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between Moduli space#Fine Moduli Spaces, fine and Moduli space#Coarse Moduli Spaces, coarse moduli spaces for the same moduli problem. The most basic problem is that of moduli of smooth morphism, smooth complete variety, complete curves of a fixed Genus (mathematics), genus. Over the field (mathematics), field of complex numbers these correspond precisely to compact Riemann surfaces of the given genus, for which Bernhard Riemann proved the first results about moduli spaces, in particular their dimensions ("number of parameters on which the complex structure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Scheme
In algebraic geometry, an algebraic variety or scheme ''X'' is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety ''X'' (understood to be irreducible) is normal if and only if the ring ''O''(''X'') of regular functions on ''X'' is an integrally closed domain. A variety ''X'' over a field is normal if and only if every finite birational morphism from any variety ''Y'' to ''X'' is an isomorphism. Normal varieties were introduced by . Geometric and algebraic interpretations of normality A morphism of varieties is finite if the inverse image of every point is finite and the morphism is proper. A morphism of varieties is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve ''X'' in the affine plane ''A''2 defined by ''x''2 = ''y''3 is not normal, because there is a finite birational morphism ''A''1 → ''X'' (namely, ''t'' maps to (''t''3, ''t''2) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Scheme
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 dimen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Dual Numbers
In algebra, the dual numbers are a hypercomplex number, hypercomplex number system first introduced in the 19th century. They are expression (mathematics), expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0. Dual numbers can be added component-wise, and multiplied by the formula : (a+b\varepsilon)(c+d\varepsilon) = ac + (ad+bc)\varepsilon, which follows from the property and the fact that multiplication is a bilinear operation. The dual numbers form a commutative algebra (structure), commutative algebra of dimension (linear algebra), dimension two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent element, nilpotent elements. History Dual numbers were introduced in 1873 by William Kingdon Clifford, William Clifford, and were used at the beginning of the twentieth century by the German mathematician Eduard Study, who used them ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deformation (algebraic Geometry)
Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Deformation (meteorology), a measure of the rate at which the shapes of clouds and other fluid bodies change. * Deformation (mathematics), the study of conditions leading to slightly different solutions of mathematical equations, models and problems. * Deformation (volcanology), a measure of the rate at which the shapes of volcanoes change. * Deformation (biology), a harmful mutation or other deformation in an organism. See also * Deformity (medicine), a major difference in the shape of a body part or organ compared to its common or average shape. * Plasticity (physics) In physics and materials science, plasticity, also known as plastic deformation, is the ability of a solid material to undergo permanent Deformation (engineering), def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Module
In algebra, a flat module over a ring ''R'' is an ''R''-module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Flatness was introduced by in his paper '' Géometrie Algébrique et Géométrie Analytique''. See also flat morphism. Definition A module over a ring is ''flat'' if the following condition is satisfied: for every injective linear map \varphi: K \to L of -modules, the map :\varphi \otimes_R M: K \otimes_R M \to L \otimes_R M is also injective, where \varphi \otimes_R M is the map induced by k \otimes m \mapsto \varphi(k) \otimes m. For this definition, it is enough to restrict the injections \varphi to the inclusions of finitely generated ideals into . Equivalently, an -module is flat if the tensor product with is an exact functor; that is if, for every short exact sequence of - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deformation Theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces. Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of ''isolated solutions'', in that varying a solution may not be possible, ''or'' does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. For example, in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Graded Lie Algebra
In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible. Such objects have applications in deformation theory and rational homotopy theory. Definition A differential graded Lie algebra is a graded vector space L = \bigoplus L_i over a field of characteristic zero together with a bilinear map cdot,\cdotcolon L_i \otimes L_j \to L_ and a differential d: L_i \to L_ satisfying : ,y= (-1)^ ,x the graded Jacobi identity: :(-1)^ ,z.html"_;"title=",[y,z">, ,[y,z_+(-1)^[y,[z,x.html"_;"title=",z.html"_;"title=",[y,z">,[y,z_+(-1)^[y,[z,x">,z.html"_;"title=",[y,z">,[y,z_+(-1)^[y,[z,x_+(-1)^[z,[x,y">,z">,[y,z_+(-1)^[y,[z,x.html"_;"title=",z.html"_;"title=",[y,z">,[y,z_+(-1)^[y,[z,x">,z.html"_;"title=",[y,z">,[y,z_+(-1)^[y,[z,x_+(-1)^[z,[x,y_=_0, and_the_graded_product_rule.html" ;"title=",z_+(-1)^[y,[z,x_+(-1)^[z,[x,y.html" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kodaira–Spencer Map
In mathematics, the Kodaira–Spencer map, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold ''X'', taking a tangent space of a point of the deformation space to the first cohomology group of the sheaf of vector fields on ''X''. Definition Historical motivation The Kodaira–Spencer map was originally constructed in the setting of complex manifolds. Given a complex analytic manifold M with charts U_i and biholomorphic maps f_ sending z_k \to z_j = (z_j^1,\ldots, z_j^n) gluing the charts together, the idea of deformation theory is to replace these transition maps f_(z_k) by parametrized transition maps f_(z_k, t_1,\ldots, t_k) over some base B (which could be a real manifold) with coordinates t_1,\ldots, t_k, such that f_(z_k, 0,\ldots, 0) = f_(z_k). This means the parameters t_i deform the complex structure of the original complex manifold M. Then, these functions must also satisfy a cocycle conditi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Splitting
In mathematics, a Frobenius splitting, introduced by , is a splitting of the injective morphism O''X''→F*O''X'' from a structure sheaf O''X'' of a characteristic ''p'' > 0 variety ''X'' to its image F*O''X'' under the Frobenius endomorphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism ma ... F*. give a detailed discussion of Frobenius splittings. A fundamental property of Frobenius-split projective schemes ''X'' is that the higher cohomology ''H''''i''(''X'',''L'') (''i'' > 0) of ample line bundles ''L'' vanishes. References * * External linksConferenceon Frobenius splitting in algebraic geometry, commutative algebra, and representation theory at Michigan, 2010. Algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Effective Cartier Divisor
In algebraic geometry, a relative effective Cartier divisor is roughly a family of effective Cartier divisors. Precisely, an effective Cartier divisor in a scheme ''X'' over a ring ''R'' is a closed subscheme ''D'' of ''X'' that (1) is flat over ''R'' and (2) the ideal sheaf I(D) of ''D'' is locally free of rank one (i.e., invertible sheaf). Equivalently, a closed subscheme ''D'' of ''X'' is an effective Cartier divisor if there is an open affine cover U_i = \operatorname A_i of ''X'' and nonzerodivisors f_i \in A_i such that the intersection D \cap U_i is given by the equation f_i = 0 (called local equations) and A / f_i A is flat over ''R'' and such that they are compatible. An effective Cartier divisor as the zero-locus of a section of a line bundle Let ''L'' be a line bundle on ''X'' and ''s'' a section of it such that s: \mathcal_X \hookrightarrow L (in other words, ''s'' is a \mathcal_X(U)- regular element for any open subset ''U''.) Choose some open cover \ of ''X'' such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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