|
Complete Algebraic Curve
In algebraic geometry, a complete algebraic curve is an algebraic curve that is complete variety, complete as an algebraic variety. A projective curve, a dimension-one projective variety, is a complete curve. A complete curve (over an algebraically closed field) is projective. Because of this, over an algebraically closed field, the terms "projective curve" and "complete curve" are usually used interchangeably. Over a more general base scheme, the distinction still matters. A curve in \mathbb^3 is called an (algebraic) space curve, while a curve in \mathbb^2 is called a plane curve. By means of a projection from a point, any smooth projective curve can be embedded into \mathbb^3; thus, up to a projection, every (smooth) curve is a space curve. Up to a birational morphism, every (smooth) curve can be embedded into \mathbb^2 as a nodal curve. Riemann's existence theorem says that the category of compact Riemann surfaces is equivalent to that of smooth projective curves over the comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenization of a polynomial, homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse function, inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. If the defining polynomial of a plane algebraic curve is irreducible polynomial, irreducible, then one has an ''irreducible plane algebraic curve''. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its ''Irreduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Max Noether's Theorem
In algebraic geometry, Max Noether's theorem may refer to the results of Max Noether: * Several closely related results of Max Noether on canonical curves * AF+BG theorem, or Max Noether's fundamental theorem, a result on algebraic curves in the projective plane, on the residual sets of intersections * Max Noether's theorem on curves lying on algebraic surfaces, which are hypersurfaces in ''P''3, or more generally complete intersections * Noether's theorem on rationality for surfaces * Max Noether theorem on the generation of the Cremona group by quadratic transformations See also *Noether's theorem, usually referring to a result derived from work of Max's daughter Emmy Noether *Noether inequality * Special divisor *Hirzebruch–Riemann–Roch theorem In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Product
Symmetric product may refer to: * The product operation of a symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ... * The symmetric product of tensors * The symmetric product of an algebraic curve * The Symmetric product (topology), \operatorname^n(X) or infinite symmetric product \operatorname^\infty(X) of a space ''X'' in algebraic topology {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CW Complex
In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. (open access) CW complexes have better categorical properties than simplicial complexes, but still retain a combinatorial nature that allows for computation (often with a much smaller complex). The C in CW stands for "closure-finite", and the W for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces \emptyset = X_ \subset X_0 \subset X_1 \subset \cdots such that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to the open k- bal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Graph (algebraic Curve)
In the mathematical discipline of graph theory, the dual graph of a planar graph is a graph that has a vertex for each face of . The dual graph has an edge for each pair of faces in that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge of has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of . The definition of the dual depends on the choice of embedding of the graph , so it is a property of plane graphs (graphs that are already embedded in the plane) rather than planar graphs (graphs that may be embedded but for which the embedding is not yet known). For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph. Historically, the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a topologica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Variety
In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian variety. Introduction The Jacobian variety is named after Carl Gustav Jacobi, who proved the complete version of the Abel–Jacobi theorem, making the injectivity statement of Niels Abel into an isomorphism. It is a principally polarized abelian variety, of dimension ''g'', and hence, over the complex numbers, it is a complex torus. If ''p'' is a point of ''C'', then the curve ''C'' can be mapped to a subvariety of ''J'' with the given point ''p'' mapping to the identity of ''J'', and ''C'' generates ''J'' as a group. Construction for complex curves Over the complex numbers, the Jacobian variety can be realized as the quotient space ''V''/''L'', where ''V'' is the dual of the vector space of all global holomorphic differenti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Sheaf Sequence
In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let ''M'' be a complex manifold, and write ''O''''M'' for the sheaf of holomorphic functions on ''M''. Let ''O''''M''* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelian groups. The exponential function gives a sheaf homomorphism :\exp : \mathcal O_M \to \mathcal O_M^*, because for a holomorphic function ''f'', exp(''f'') is a non-vanishing holomorphic function, and exp(''f'' + ''g'') = exp(''f'')exp(''g''). Its kernel is the sheaf 2Ï€''i''Z of locally constant functions on ''M'' taking the values 2Ï€''in'', with ''n'' an integer. The exponential sheaf sequence is therefore :0\to 2\pi i\,\mathbb Z \to \mathcal O_M\to\mathcal O_M^*\to 0. The exponential mapping here is not always a surjective map on sections; this can be seen for example when ''M'' is a punctured disk in the complex plane. ... [...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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Roch Theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus ''g'', in a way that can be carried over into purely algebraic settings. Initially proved as Riemann's inequality by , the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student . It was later generalized to algebraic curves, to higher-dimensional varieties and beyond. Preliminary notions A Riemann surface X is a topological space that is locally homeomorphic to an open subset of \Complex, the set of complex numbers. In addition, the transition maps between these open subsets are required to be holomorphic. The latter condition allows one to transfer the notions and methods of c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Curve
In algebraic geometry, a stable curve is an algebraic curve that is asymptotically stable in the sense of geometric invariant theory. This is equivalent to the condition that it is a Complete variety, complete connected curve whose only singularities are ordinary double points and whose automorphism group is finite. The condition that the automorphism group is finite can be replaced by the condition that it is not of arithmetic genus one and every non-singular rational curve, rational component meets the other components in at least 3 points . A semi-stable curve is one satisfying similar conditions, except that the automorphism group is allowed to be reductive group, reductive rather than finite (or equivalently its connected component may be a algebraic torus, torus). Alternatively the condition that non-singular rational components meet the other components in at least three points is replaced by the condition that they meet in at least two points. Similarly a curve with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Curve
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the nth exterior power of the cotangent bundle \Omega on V. Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle T^*V. Equivalently, it is the line bundle of holomorphic n-forms on V. This is the dualising object for Serre duality on V. It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor K on V giving rise to the canonical bundle — it is an equivalence class for linear equivalence on V, and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −K with K canonical. The anticanonical bundle is the corresponding inverse bundle \omega^. When the anticanonical bundle of V is ample, V is called a Fano variety. The adjunction formula Suppose that X is a smooth variety and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
étale Fundamental Group
The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces. Topological analogue/informal discussion In algebraic topology, the fundamental group \pi_1(X,x) of a pointed topological space (X, x) is defined as the group of homotopy classes of loops based at x. This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the Zariski topology. In the classification of covering spaces, it is shown that the fundamental group is exactly the group of deck transformations of the universal covering space. This is more promising: finite étale morphisms of algebraic varieties are the appropriate analogue of covering spaces of topological spaces. Unfortunately, an algebraic variety X often fails to have a "universal cover" that is finite over ''X'', so one must consider the entire category of finite étale coverings of ''X''. One ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
