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Proj Construction
In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functorial, is a fundamental tool in scheme theory. In this article, all rings will be assumed to be commutative and with identity. Proj of a graded ring Proj as a set Let S be a graded ring, whereS = \bigoplus_ S_iis the direct sum decomposition associated with the gradation. The irrelevant ideal of S is the ideal of elements of positive degreeS_+ = \bigoplus_ S_i .We say an ideal is homogeneous if it is generated by homogeneous elements. Then, as a set,\operatorname S = \. For brevity we will sometimes write X for \operatorname S. Proj as a topological space We may define a topology, called the Zariski topology, on \operatorname S by defining the closed sets to be those of the form :V(a) = \, where a is a homogeneous ideal of S. As in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Quintic Threefold
In mathematics, a Fermat quintic threefold is a special quintic threefold, in other words a degree 5, dimension 3 hypersurface in 4-dimensional complex projective space, given by the equation :V^5+W^5+X^5+Y^5+Z^5=0. This threefold, so named after Pierre de Fermat, is a Calabi–Yau manifold. The Hodge diamond of a non-singular quintic 3-fold is Rational curves conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. The Fermat quintic threefold is not generic in this sense, and showed that its lines are contained in 50 1-dimensional families of the form : (x : -\zeta x : ay : by : cy) for \zeta^5=1 and a^5+b^5+c^5=0. There are 375 lines in more than one family, of the form : (x : -\zeta x : y :-\eta y :0) for fifth roots of unity \zeta and \eta. References * * *{{Citation , last1=Cox , first1=David A. , authorlink1= David A. Cox , last2=Katz , first2=Sheldon , authorlink2= Sheldon Katz , title=Mirror symmetry and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation :x_1^2+x_2^2+\cdots+x_n^2-1=0 defines an algebraic hypersurface of dimension in the Euclidean space of dimension . This hypersurface is also a smooth manifold, and is called a hypersphere or an -sphere. Smooth hypersurface A hypersurface that is a smooth manifold is called a ''smooth hypersurface''. In , a smooth hypersurface is orienta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Ideal
In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let \mathcal(x_1,\ldots,x_n) denote the ring of smooth functions in n variables and f a function in the ring. The Jacobian ideal of f is : J_f := \left\langle \frac, \ldots, \frac \right\rangle. Relation to deformation theory In deformation theory, the deformations of a hypersurface given by a polynomial f is classified by the ring \frac This is shown using the Kodaira–Spencer map. Relation to Hodge theory In Hodge theory, there are objects called real Hodge structures which are the data of a real vector space H_\mathbb and an increasing filtration F^\bullet of H_\mathbb = H_\mathbb\otimes_\mathbb satisfying a list of compatibility structures. For a smooth projective variety X there is a canonical Hodge structure. Statement for degree d hypersurfaces In the special case X is defined by a homogeneous degree d polynomial f \in \Gamma(\mathbb^,\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Morphism
In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if *(i) it is locally of finite presentation *(ii) it is flat, and *(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular. (iii) means that each geometric fiber of ''f'' is a nonsingular variety (if it is separated). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties. If ''S'' is the spectrum of an algebraically closed field and ''f'' is of finite type, then one recovers the definition of a nonsingular variety. Equivalent definitions There are many equivalent definitions of a smooth morphism. Let f: X \to S be locally of finite presentation. Then the following are equivalent. # ''f'' is smooth. # ''f'' is formally smooth (see below). # ''f'' is flat and the sheaf of relative differentials \Omega_ is locally free of rank equal to the relative dimension of X/S. # For any x \in X, there exists a neighborhood \operatornameB of x and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Ringed Space
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of continuous (scalar-valued) functions on open subsets. Among ringed spaces, especially important and prominent is a locally ringed space: a ringed space in which the analogy between the stalk at a point and the ring of germs of functions at a point is valid. Ringed spaces appear in analysis as well as complex algebraic geometry and the scheme theory of algebraic geometry. Note: In the definition of a ringed space, most expositions tend to restrict the rings to be commutative rings, including Hartshorne and Wikipedia. "Éléments de géométrie algébrique", on the other hand, does not impose the commutativity assumption, although the book mostly considers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Global Sections Functor
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morphisms) from one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible Sheaf
In mathematics, an invertible sheaf is a coherent sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O''''X''-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties. Definition An invertible sheaf is a locally free sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O''''X''-modules, that is, we have :S \otimes T\ isomorphic to ''O''''X'', which acts as identity element for the tensor product. The most significant cases are those coming from algebraic geometry and complex geometry. For spaces such as (locally) Noetherian schemes or complex manifolds, one can actually replace 'locally free' by 'coherent' in the definition. The invertible sheaves in those theories are in effect the line bundles appropriately formulat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003. Biography Personal life Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil. The French mathematician Denis S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
