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Linear Series
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the form of a ''linear system'' of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisors ''D'' on a general scheme or even a ringed space (''X'', ''O''''X''). Linear system of dimension 1, 2, or 3 are called a pencil, a net, or a web, respectively. A map determined by a linear system is sometimes called the Kodaira map. Definition Given the fundamental idea of a rational function on a general variety X, or in other words of a function f in the function field of X, f \in k(X), divisors D,E \in \text(X) are linearly equivalent divisors if :D = E + (f)\ where (f) denotes the divisor of zeroes and poles of the function f. Note ... [...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] |
Holomorphic Line Bundle
In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle. By Serre's GAGA, the category of holomorphic vector bundles on a smooth complex projective variety ''X'' (viewed as a complex manifold) is equivalent to the category of algebraic vector bundles (i.e., locally free sheaves of finite rank) on ''X''. Definition through trivialization Specifically, one requires that the trivialization maps :\phi_U : \pi^(U) \to U \times \mathbf^k are biholomorphic maps. This is equivalent to requiring that the transition functions :t_ : U\cap V \to \mathrm_k(\mathbf) are holomorphic maps. The holomorphic structure on the tangent bundle of a complex manifold is guaranteed by t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold. The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two). Many results were obtained, however, in the Italian school of algebraic geometry, and are up to 100 years old. Classification by the Kodaira dimension In the case of dimension one varieties are classified by only the topological genus, but dimension two, the difference between the arithmetic genus p_a and the geometric genus p_g turns to be important because we cannot distinguish birationally only the topological genus. Then we introduce the irregularity for the classification of them. A summary of the results (in det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Abelian Group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors. The elements of a free abelian group with basis B may be described in several equivalent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of module (mathematics), modules and Syzygy (mathematics), syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariant (mathematics), invariants of ring (mathematics), rings, modules, topological spaces, and other 'tan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Italian School Of Algebraic Geometry
In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 to 40 leading mathematicians who made major contributions, about half of those being Italian. The leadership fell to the group in Rome of Guido Castelnuovo, Federigo Enriques and Francesco Severi, who were involved in some of the deepest discoveries, as well as setting the style. Algebraic surfaces The emphasis on algebraic surfaces—algebraic varieties of dimension two—followed on from an essentially complete geometric theory of algebraic curves (dimension 1). The position in around 1870 was that the curve theory had incorporated with Brill–Noether theory the Riemann–Roch theorem in all its refinements (via the detailed geometry of the theta-divisor). The classification of algebraic surfaces was a bold and successful att ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birational Geometry
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles. Birational maps Rational maps A rational map from one variety (understood to be irreducible) X to another variety Y, written as a dashed arrow , is defined as a morphism from a nonempty open subset U \subset X to Y. By definition of the Zariski topology used in algebraic geometry, a nonempty open subset U is always dense in X, in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions. Birational maps A birational map from ''X'' to ''Y'' is a rational map such that there is a rational map inverse to ''f''. A birational map induces an isomorphism from a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley–Bacharach Theorem
In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane . The original form states: :Assume that two cubics and in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field. Then every cubic that passes through any eight of the points also passes through the ninth point. A more intrinsic form of the Cayley–Bacharach theorem reads as follows: :Every cubic curve over an algebraically closed field that passes through a given set of eight points also passes through (counting multiplicities) a ninth point which depends only on . A related result on conics was first proved by the French geometer Michel Chasles and later generalized to cubics by Arthur Cayley and Isaak Bacharach. Details If seven of the points lie on a conic, then the ninth point can be chosen on that conic, since will always contain the whole conic on account of Bézout's the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Bundle
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Definition Riemannian manifold Let (M,g) be a Riemannian manifold, and S \subset M a Riemannian submanifold. Define, for a given p \in S, a vector n \in \mathrm_p M to be ''normal'' to S whenever g(n,v)=0 for all v\in \mathrm_p S (so that n is orthogonal to \mathrm_p S). The set \mathrm_p S of all such n is then called the ''normal space'' to S at p. Just as the total space of the tangent bundle to a manifold is constructed from all tangent spaces to the manifold, the total space of the normal bundle \mathrm S to S is defined as :\mathrmS := \coprod_ \mathrm_p S. The conormal bundle is defined as the dual bundle to the normal bundle. It can be realised naturally as a sub-bundle of the cotangent bundle. General definition More abstractly, given an immersion i: N \to M (for instance an embeddin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonal Curve
In mathematics, the gonality of an algebraic curve ''C'' is defined as the lowest degree of a nonconstant rational map from ''C'' to the projective line. In more algebraic terms, if ''C'' is defined over the field ''K'' and ''K''(''C'') denotes the function field of ''C'', then the gonality is the minimum value taken by the degrees of field extensions :''K''(''C'')/''K''(''f'') of the function field over its subfields generated by single functions ''f''. If ''K'' is algebraically closed, then the gonality is 1 precisely for curves of genus 0. The gonality is 2 for curves of genus 1 (elliptic curves) and for hyperelliptic curves (this includes all curves of genus 2). For genus ''g'' ≥ 3 it is no longer the case that the genus determines the gonality. The gonality of the generic curve of genus ''g'' is the floor function of :(''g'' + 3)/2. Trigonal curves are those with gonality 3, and this case gave rise to the name in general. Trigonal curves include the Picard curves, of gen ... [...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 complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
