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Toric Varieties
In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be normal. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special, but still quite general class of toric varieties, this information is also encoded in a polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space. Toric varieties from tori The original motivation to study toric varieties was to study torus embeddings. Given the algebrai ... [...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] |
Resolution Of Singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a proper birational map ''W''→''V''. For varieties over fields of characteristic 0 this was proved in Hironaka (1964), while for varieties over fields of characteristic ''p'' it is an open problem in dimensions at least 4. Definitions Originally the problem of resolution of singularities was to find a nonsingular model for the function field of a variety ''X'', in other words a complete non-singular variety ''X′'' with the same function field. In practice it is more convenient to ask for a different condition as follows: a variety ''X'' has a resolution of singularities if we can find a non-singular variety ''X′'' and a proper birational map from ''X′'' to ''X''. The condition that the map is proper is needed to exclude trivial solutions, such as taking ''X′'' to be the subvariety of non- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Stack
In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack. The notion is of fundamental importance in the study of stacks: a stack that arises in nature is often either a quotient stack itself or admits a stratification by quotient stacks (e.g., a Deligne–Mumford stack.) A quotient stack is also used to construct other stacks like classifying stacks. Definition A quotient stack is defined as follows. Let ''G'' be an affine smooth group scheme over a scheme ''S'' and ''X'' an ''S''-scheme on which ''G'' acts. Let the quotient stack /G/math> be the category over the category of ''S''-schemes: *an object over ''T'' is a principal ''G''-bundle P\to T together with equivariant map P\to X; *an arrow from P\to T to P'\to T' is a bundle map (i.e., forms a commutative diagram) that is compatible with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GIT Quotient
In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme X = \operatorname A with an action by a group scheme ''G'' is the affine scheme \operatorname(A^G), the prime spectrum of the ring of invariants of ''A'', and is denoted by X /\!/ G. A GIT quotient is a categorical quotient: any invariant morphism uniquely factors through it. Taking Proj (of a graded ring) instead of \operatorname, one obtains a projective GIT quotient (which is a quotient of the set of semistable points.) A GIT quotient is a categorical quotient of the locus of semistable points; i.e., "the" quotient of the semistable locus. Since the categorical quotient is unique, if there is a geometric quotient, then the two notions coincide: for example, one has :G / H = G /\!/ H = \operatorname\!\big(k H\big) for an algebraic group ''G'' over a field ''k'' and closed subgroup ''H''. If ''X'' is a complex smooth projective variety and if ''G'' is a reductive co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toric Stack
In algebraic geometry, a toric stack is a stacky generalization of a toric variety. More precisely, a toric stack is obtained by replacing in the construction of a toric variety a step of taking GIT quotients with that of taking quotient stacks. Consequently, a toric variety is a coarse approximation of a toric stack. A toric orbifold is an example of a toric stack. See also *Stanley–Reisner ring In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ... References * * * Algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toric Ideal
In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field. A toric ideal is an ideal generated by differences of monomials (provided the ideal is a prime ideal). An affine or projective algebraic variety defined by a toric ideal or a homogeneous toric ideal is an affine or projective toric variety, possibly non-normal. Definitions and Properties Let \mathbb be a field and R = \mathbb /math> be the polynomial ring over \mathbb with ''n'' variables x = x_1, x_2, \dotsc, x_n. A monomial in R is a product x^ = x_1^ x_2^ \cdots x_n^ for an ''n''-tuple \alpha = (\alpha_1, \alpha_2, \dotsc, \alpha_n) \in \mathbb^n of nonnegative integers. The following three conditions are equivalent for an ideal I \subseteq R: # I is generated by monomials, # If f = \sum_ c_ x^ \in I , then x^ \in I , provided that c_ is nonzero. # I is torus fixed, i.e, given (c_1, c_2, \dotsc, c_n) \in (\mathbb^*)^ , then I is fixed under the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gordan's Lemma
Gordan's lemma is a lemma in convex geometry and algebraic geometry. It can be stated in several ways. * Let A be a matrix of integers. Let M be the set of non-negative integer solutions of A \cdot x = 0. Then there exists a finite subset of vectors in M, such that every element of M is a linear combination of these vectors with non-negative integer coefficients. * The semigroup of integral points in a rational convex polyhedral cone is finitely generated. * An affine toric variety is an algebraic variety (this follows from the fact that the prime spectrum of the semigroup algebra of such a semigroup is, by definition, an affine toric variety). The lemma is named after the mathematician Paul Gordan (1837–1912). Some authors have misspelled it as "Gordon's lemma". Proofs There are topological and algebraic proofs. Topological proof Let \sigma be the dual cone of the given rational polyhedral cone. Let u_1, \dots, u_r be integral vectors so that \sigma = \. Then the u_i's g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Notices Of The American Mathematical Society
''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume appeared in 1953. Each issue of the magazine since January 1995 is available in its entirety on the journal web site. Articles are peer-reviewed by an editorial board of mathematical experts. Since 2019, the editor-in-chief is Erica Flapan. The cover regularly features mathematical visualization Mathematical phenomena can be understood and explored via visualization. Classically this consisted of two-dimensional drawings or building three-dimensional models (particularly plaster models in the 19th and early 20th century), while today it ...s. The ''Notices'' is self-described to be the world's most widely read mathematical journal. As the membership journal of the American Mathematical Society, the ''Notices'' is sent to the approximately 30,000 AMS members worldwide, one-third of whom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, ''The Daily Princetonian'', and later added book publishing to it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mirror Symmetry (string Theory)
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory. Early cases of mirror symmetry were discovered by physicists. Mathematicians became interested in this relationship around 1990 when Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that it could be used as a tool in enumerative geometry, a branch of mathematics concerned with counting the number of solutions to geometric questions. Candelas and his collaborators showed that mirror symmetry could be used to count rational curves on a Calabi–Yau manifold, thus solving a longstanding problem. Although the original approach to mirror symmetry was based on physical ideas that were not understood in a mathematically precise way, some of its mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
U(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. The circle group forms a subgroup of \mathbb C^\times, the multiplicative group of all nonzero complex numbers. Since \mathbb C^\times is abelian, it follows that \mathbb T is as well. A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure \theta: \theta \mapsto z = e^ = \cos\theta + i\sin\theta. This is the exponential map for the circle group. The circle group plays a central role in Pontryagin duality and in the theory of Lie groups. The notation \mathbb T for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, \mathbb T^n (the direct product of \mathbb T with it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
