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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] |
Functorial Resolution Of Singularities
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a linguistic context; see function word. Definition Let ''C'' and ''D'' be categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F(Y) in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excellent Ring
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities can be solved; showed this in characteristic (algebra), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pinch Point (mathematics)
image:Whitney_unbrella.png, frame, Section of the Whitney umbrella, an example of pinch point singularity. In geometry, a pinch point or cuspidal point is a type of Singular point of an algebraic variety, singular point on an algebraic surface. The equation for the surface near a pinch point may be put in the form : f(u,v,w) = u^2 - vw^2 + [4] \, where [4] denotes Term (logic), terms of Degree of a monomial, degree 4 or more and v is not a square in the ring of functions. For example the surface 1-2x+x^2-yz^2=0 near the point (1,0,0), meaning in coordinates vanishing at that point, has the form above. In fact, if u=1-x, v=y and w=z then is a system of coordinates vanishing at (1,0,0) then 1-2x+x^2-yz^2=(1-x)^2-yz^2=u^2-vw^2 is written in the canonical form. The simplest example of a pinch point is the hypersurface defined by the equation u^2-vw^2=0 called Whitney umbrella. The pinch point (in this case the origin) is a limit of Normal crossing divisor, normal crossings sing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Umbrella
frame, Section of the surface In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface placed in three dimensions. It is the union of all straight lines that pass through points of a fixed parabola and are perpendicular to a fixed straight line which is parallel to the axis of the parabola and lies on its perpendicular bisecting plane. Formulas Whitney's umbrella can be given by the parametric equations in Cartesian coordinates : \left\{\begin{align} x(u, v) &= uv, \\ y(u, v) &= u, \\ z(u, v) &= v^2, \end{align}\right. where the parameters ''u'' and ''v'' range over the real numbers. It is also given by the implicit equation : x^2 - y^2 z = 0. This formula also includes the negative ''z'' axis (which is called the ''handle'' of the umbrella). Properties Whitney's umbrella is a ruled surface and a right conoid. It is important in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Unbrella
Whitney may refer to: Film and television * Whitney (2015 film), ''Whitney'' (2015 film), a Whitney Houston biopic starring Yaya DaCosta * Whitney (2018 film), ''Whitney'' (2018 film), a documentary about Whitney Houston * Whitney (TV series), ''Whitney'' (TV series), an American sitcom that premiered in 2011 Firearms *Whitney Wolverine, a semi-automatic, .22 LR caliber pistol *Whitney revolver, a gun carried by Assassination of Abraham Lincoln#Powell attacks Seward, Powell when he attempted to assassinate Secretary of State William Seward Music * Whitney Houston, sometimes eponymously known as 'Whitney' ** Whitney (album), ''Whitney'' (album), an album by Whitney Houston * Whitney (band), an American rock band Places Canada * Whitney, Ontario United Kingdom * Witney, Oxfordshire ** Witney (UK Parliament constituency), a constituency for the House of Commons * Whitney-on-Wye, Herefordshire United States * Whitney, Alabama * Whitney, California, a community in Placer Count ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Samuel Function
In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel,H. Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I. Ann. of Math. 2nd Ser., Vol. 79, No. 1. (Jan., 1964), pp. 109-203. of a nonzero finitely generated module M over a commutative Noetherian local ring A and a primary ideal I of A is the map \chi_^:\mathbb\rightarrow\mathbb such that, for all n\in\mathbb, :\chi_^(n)=\ell(M/I^M) where \ell denotes the length over A. It is related to the Hilbert function of the associated graded module \operatorname_I(M) by the identity : \chi_M^I (n)=\sum_^n H(\operatorname_I(M),i). For sufficiently large n, it coincides with a polynomial function of degree equal to \dim(\operatorname_I(M)), often called the Hilbert-Samuel polynomial (or Hilbert polynomial).Atiyah, M. F. and MacDonald, I. G. ''Introduction to Commutative Algebra''. Reading, MA: Addison–Wesley, 1969. Examples For the ring of form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exceptional Divisor
In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map :f: X \rightarrow Y of varieties is a kind of 'large' subvariety of X which is 'crushed' by f, in a certain definite sense. More strictly, ''f'' has an associated exceptional locus which describes how it identifies nearby points in codimension one, and the exceptional divisor is an appropriate algebraic construction whose support is the exceptional locus. The same ideas can be found in the theory of holomorphic mappings of complex manifolds. More precisely, suppose that :f: X \rightarrow Y is a regular map of varieties which is birational (that is, it is an isomorphism between open subsets of X and Y). A codimension-1 subvariety Z \subset X is said to be ''exceptional'' if f(Z) has codimension at least 2 as a subvariety of Y. One may then define the ''exceptional divisor'' of f to be :\sum_i Z_i \in Div(X), where the sum is over all exceptional subvarieties of f, and is an element of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Sheaf
In algebraic geometry and other areas of mathematics, an ideal sheaf (or sheaf of ideals) is the global analogue of an ideal in a ring. The ideal sheaves on a geometric object are closely connected to its subspaces. Definition Let ''X'' be a topological space and ''A'' a sheaf of rings on ''X''. (In other words, (''X'', ''A'') is a ringed space.) An ideal sheaf ''J'' in ''A'' is a subobject of ''A'' in the category of sheaves of ''A''-modules, i.e., a subsheaf of ''A'' viewed as a sheaf of abelian groups such that : Γ(''U'', ''A'') · Γ(''U'', ''J'') ⊆ Γ(''U'', ''J'') for all open subsets ''U'' of ''X''. In other words, ''J'' is a sheaf of ''A''-submodules of ''A''. General properties * If ''f'': ''A'' → ''B'' is a homomorphism between two sheaves of rings on the same space ''X'', the kernel of ''f'' is an ideal sheaf in ''A''. * Conversely, for any ideal sheaf ''J'' in a sheaf of rings ''A'', there is a natural structure of a sheaf of rings on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-excellent
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities can be solved; showed this in characteristic 0, but the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski–Riemann Space
In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring ''k'' of a field ''K'' is a locally ringed space whose points are valuation rings containing ''k'' and contained in ''K''. They generalize the Riemann surface of a complex curve. Zariski–Riemann spaces were introduced by who (rather confusingly) called them Riemann manifolds or Riemann surfaces. They were named Zariski–Riemann spaces after Oscar Zariski and Bernhard Riemann by who used them to show that algebraic varieties can be embedded in complete ones. Local uniformization (proved in characteristic 0 by Zariski) can be interpreted as saying that the Zariski–Riemann space of a variety is nonsingular in some sense, so is a sort of rather weak resolution of singularities. This does not solve the problem of resolution of singularities because in dimensions greater than 1 the Zariski–Riemann space is not locally affine and in particular is not a scheme. Definition The Zariski–Riemann ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excellent Scheme
Excellent may refer to: * Excellent (album), ''Excellent'' (album), by Propaganda, 2012 * "Excellent", a song by Sunday Service Choir from the 2019 album ''Jesus Is Born'' * "Excellent", a catchphrase of Mr. Burns in the cartoon ''The Simpsons'' * "Excellent!," a catchphrase of the title characters in the ''Bill & Ted'' movie franchise * , a ship and a shore establishment of the Royal Navy See also * Excellence (other) * Excellent ring, in commutative algebra * Most Excellent Majesty, a form of address in the United Kingdom {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
