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Relative Effective Cartier Divisor
In algebraic geometry, a relative effective Cartier divisor is roughly a family of effective Cartier divisors. Precisely, an effective Cartier divisor in a scheme ''X'' over a ring ''R'' is a closed subscheme ''D'' of ''X'' that (1) is flat over ''R'' and (2) the ideal sheaf I(D) of ''D'' is locally free of rank one (i.e., invertible sheaf). Equivalently, a closed subscheme ''D'' of ''X'' is an effective Cartier divisor if there is an open affine cover U_i = \operatorname A_i of ''X'' and nonzerodivisors f_i \in A_i such that the intersection D \cap U_i is given by the equation f_i = 0 (called local equations) and A / f_i A is flat over ''R'' and such that they are compatible. An effective Cartier divisor as the zero-locus of a section of a line bundle Let ''L'' be a line bundle on ''X'' and ''s'' a section of it such that s: \mathcal_X \hookrightarrow L (in other words, ''s'' is a \mathcal_X(U)- regular element for any open subset ''U''.) Choose some open cover \ of ''X'' such ... [...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] |
Effective Cartier Divisor
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-''r'' subvariety need not be definable by only ''r'' equations when ''r'' is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this property, much of algebraic geometry studies an arbitrary variety by analysing its codimension-1 subvarieti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e., :f_P\colon \mathcal_ \to \mathcal_ is a flat map for all ''P'' in ''X''. A map of rings A\to B is called flat if it is a homomorphism that makes ''B'' a flat ''A''-module. A morphism of schemes is called faithfully flat if it is both surjective and flat. Two basic intuitions regarding flat morphisms are: *flatness is a generic property; and *the failure of flatness occurs on the jumping set of the morphism. The first of these comes from commutative algebra: subject to some finiteness conditions on ''f'', it can be shown that there is a non-empty open subscheme Y' of ''Y'', such that ''f'' restricted to ''Y''′ is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of the fiber product of schemes, applied to ''f'' and the inclusio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero divisor if there exists a nonzero in such that . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero such that may be different from the nonzero such that ). If the ring is commutative, then the left and right zero divisors are the same. An element of a ring that is not a left zero divisor is called left regular or left cancellable. Similarly, an element of a ring that is not a right zero divisor is called right regular or right cancellable. An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regu ... [...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] |
Proper Scheme
In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field ''k'' is proper over ''k''. A scheme ''X'' of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space ''X''(C) of complex points with the classical (Euclidean) topology is compact and Hausdorff. A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite. Definition A morphism ''f'': ''X'' → ''Y'' of schemes is called universally closed if for every scheme ''Z'' with a morphism ''Z'' → ''Y'', the projection from the fiber product :X \times_Y Z \to Z is a closed map of the underlying topological spaces. A morphism of schemes is called proper if it is separated, of finite type, and universally closed ( GAII, 5.4.. One also says that ''X'' is pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism Of Finite Presentation
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as t ... [...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] |
