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Zariski's Connectedness Theorem
In algebraic geometry, Zariski's connectedness theorem (due to Oscar Zariski) says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's main theorem to the case when the morphism of varieties need not be birational. Zariski's connectedness theorem gives a rigorous version of the "principle of degeneration" introduced by Federigo Enriques, which says roughly that a limit of absolutely irreducible cycles is absolutely connected. Statement Suppose that ''f'' is a proper surjective morphism of varieties from ''X'' to ''Y'' such that the function field of ''Y'' is separably closed In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1 ... in that of ''X''. Then Zariski's connectedness theorem says that the inverse image of any norma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oscar Zariski
Oscar Zariski (April 24, 1899 – July 4, 1986) was an American mathematician. The Russian-born scientist was one of the most influential algebraic geometers of the 20th century. Education Zariski was born Oscher (also transliterated as Ascher or Osher) Zaritsky to a Jewish family (his parents were Bezalel Zaritsky and Hanna Tennenbaum) and in 1918 studied at the University of Kyiv. He left Kyiv in 1920 to study at the University of Rome where he became a disciple of the Italian school of algebraic geometry, studying with Guido Castelnuovo, Federigo Enriques and Francesco Severi. Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory, which was proposed to him by Castelnuovo. At the time of his dissertation publication, he changed his name to Oscar Zariski. Johns Hopkins University years Zariski emigrated to the United States in 1927 supported by Solomon Lefschetz. He had a position at Johns Hopkins University where he became professor in 1937. During this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski's Main Theorem
In algebraic geometry, Zariski's main theorem, proved by , is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness theorem when the two varieties are birational. Zariski's main theorem can be stated in several ways which at first sight seem to be quite different, but are in fact deeply related. Some of the variations that have been called Zariski's main theorem are as follows: *A birational morphism with finite fibers to a normal variety is an isomorphism to an open subset. *The total transform of a normal fundamental point of a birational map has positive dimension. This is essentially Zariski's original version. *The total transform of a normal point under a proper birational morphism is connected. *A generalization due to Grothendieck describes the structure of quasi-finite morphisms of schemes. Several results in commutative algebra imply the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Federigo Enriques
Abramo Giulio Umberto Federigo Enriques (5 January 1871 – 14 June 1946) was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry. Biography Enriques was born in Livorno, and brought up in Pisa, in a Sephardi Jewish family of Portuguese descent. His younger brother was zoologist Paolo Enriques who was also the father of Enzo Enriques Agnoletti and Anna Maria Enriques Agnoletti. He became a student of Guido Castelnuovo (who later became his brother-in-law after marrying his sister Elbina), and became an important member of the Italian school of algebraic geometry. He also worked on differential geometry. He collaborated with Castelnuovo, Corrado Segre and Francesco Severi. He had positions at the University of Bologna, and then the University of Rome La Sapienza. In 1931, he swore allegiance to fascism, and in 1933 he became a member of the PNF. Des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Morphism
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\to Y of schemes is called universally closed if for every scheme Z with a morphism Z\to Y, the projection from the fiber product :X \times_Y Z \to Z is a closed map of the underlying topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism Of Varieties
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and the biregular maps are the isomorphisms of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the concepts of rational and birational maps are widely used as well; they are partial functions that are defined locally by rational fractions instead of polynomials. An algebraic variety has naturally the structure of a locally ringed space; a morphism between algebraic varieties is precisely a morphism of the underlying locally ringed spaces. Definition If ''X'' and ''Y'' are closed subvarieties of \mathbb^n and \mathbb^m (so they are affine varieties ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Field Of An Algebraic Variety
In algebraic geometry, the function field of an algebraic variety ''V'' consists of objects that are interpreted as rational functions on ''V''. In classical algebraic geometry they are ratios of polynomials; in complex geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions. Definition for complex manifolds In complex geometry the objects of study are complex analytic varieties, on which we have a local notion of complex analysis, through which we may define meromorphic functions. The function field of a variety is then the set of all meromorphic functions on the variety. (Like all meromorphic functions, these take their values in \mathbb\cup\.) Together with the operations of addition and multiplication of functions, this is a field in the sense of algebra. For the Riemann sphere, which is the variety \mathbb^1 over the complex numbers, the global meromorphic fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separably Closed
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1972) pp.74-76 or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field ''K'' is unique up to an isomorphism that fixes every member of ''K''. Because of this essential uniqueness, we often speak of ''the'' algebraic closure of ''K'', rather than ''an'' algebraic closure of ''K''. The algebraic closure of a field ''K'' can be thought of as the largest algebraic extension of ''K''. To see this, note that if ''L'' is any algebraic extension of ''K'', then the algebraic closure of ''L'' is also an algebraic closure of ''K'', and so ''L'' is contained within the algebraic closure of ''K''. The algebraic closure of ''K'' is also the smallest algebraically closed fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Memoirs Of The American Mathematical Society
''Memoirs of the American Mathematical Society'' is a mathematical journal published by the American Mathematical Society and intended to carry peer-reviewed book-length research publications (monographs). As such, it has been described as having the characteristics both of a journal and of a book series. As with other journals, its publications are collected into volumes, numbering six per year until 2024 and twelve per year from 2025 on. The start of the journal was announced in 1949, and it began publication in 1950. The journal is indexed by Mathematical Reviews, Zentralblatt MATH, Science Citation Index, Research Alert, CompuMath Citation Index, and Current Contents. Its dual nature as a journal and a book series has sometimes caused inconsistencies in its indexing. Other journals from the AMS * ''Bulletin of the American Mathematical Society'' * ''Journal of the American Mathematical Society'' * ''Notices of the American Mathematical Society'' * ''Proceedings of the Ame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
