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Normalization Of An Algebraic Variety
In algebraic geometry, an algebraic variety or scheme ''X'' is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety ''X'' (understood to be irreducible) is normal if and only if the ring ''O''(''X'') of regular functions on ''X'' is an integrally closed domain. A variety ''X'' over a field is normal if and only if every finite birational morphism from any variety ''Y'' to ''X'' is an isomorphism. Normal varieties were introduced by . Geometric and algebraic interpretations of normality A morphism of varieties is finite if the inverse image of every point is finite and the morphism is proper. A morphism of varieties is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve ''X'' in the affine plane ''A''2 defined by ''x''2 = ''y''3 is not normal, because there is a finite birational morphism ''A''1 → ''X'' (namely, ''t'' maps to (''t''3, ''t''2)) ... [...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] |
Linearly Normal
In algebraic geometry, the homogeneous coordinate ring ''R'' of an algebraic variety ''V'' given as a subvariety of projective space of a given dimension ''N'' is by definition the quotient ring :''R'' = ''K'' 'X''0, ''X''1, ''X''2, ..., ''X''''N''thinsp;/''I'' where ''I'' is the homogeneous ideal defining ''V'', ''K'' is the algebraically closed field over which ''V'' is defined, and :''K'' 'X''0, ''X''1, ''X''2, ..., ''X''''N'' is the polynomial ring in ''N'' + 1 variables ''X''''i''. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice of basis (in the vector space underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra. Formulation Since ''V'' is assumed to be a variety, and so an irreducible algebraic set, the ideal ''I'' can be chosen to be a prime ideal, and so ''R'' is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbac ... [...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] |
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] |
Noether Normalization Lemma
In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field ''k'', and any finitely generated commutative ''k''-algebra ''A'', there exists a non-negative integer ''d'' and algebraically independent elements ''y''1, ''y''2, ..., ''y''''d'' in ''A'' such that ''A'' is a finitely generated module over the polynomial ring ''S'' = ''k'' 'y''1, ''y''2, ..., ''y''''d'' The integer ''d'' above is uniquely determined; it is the Krull dimension of the ring ''A''. When ''A'' is an integral domain, ''d'' is also the transcendence degree of the field of fractions of ''A'' over ''k''. The theorem has a geometric interpretation. Suppose ''A'' is integral. Let ''S'' be the coordinate ring of the ''d''-dimensional affine space \mathbb A^d_k, and let ''A'' be the coordinate ring of some other ''d''-dimensional affine variety ''X''. Then the inclusion map ''S'' → ''A'' induces a surjective f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Closure
In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a''''j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' is a root of a monic polynomial over ''A''. The set of elements of ''B'' that are integral over ''A'' is called the integral closure of ''A'' in ''B''. It is a subring of ''B'' containing ''A''. If every element of ''B'' is integral over ''A'', then we say that ''B'' is integral over ''A'', or equivalently ''B'' is an integral extension of ''A''. If ''A'', ''B'' are fields, then the notions of "integral over" and of an "integral extension" are precisely " algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial). The case of greatest interest in number theory is that of complex numbers integral over Z (e.g., \sqrt or 1+i); in this context, the integral elements are usual ... [...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] |
Reduced Scheme
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme ''S'' and a morphism an ''S''-morphism. !$@ A B C D E F G H I J K L M N O P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glossary Of Scheme Theory
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme ''S'' and a morphism an ''S''-morphism. !$@ A B C D E F G H I J K L M N O P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Normal Scroll
In mathematics, a rational normal scroll is a ruled surface of degree ''n'' in projective space of dimension ''n'' + 1. Here "rational" means birational to projective space, "scroll" is an old term for ruled surface, and "normal" refers to projective normality (not normal schemes). A non-degenerate irreducible surface of degree ''m'' – 1 in P''m'' is either a rational normal scroll or the Veronese surface. Construction In projective space of dimension ''m'' + ''n'' + 1 choose two complementary linear subspaces of dimensions ''m'' > 0 and ''n'' > 0. Choose rational normal curves in these two linear subspaces, and choose an isomorphism φ between them. Then the rational normal surface consists of all lines joining the points ''x'' and ''φ''(''x''). In the degenerate case when one of ''m'' or ''n'' is 0, the rational normal scroll becomes a cone over a rational normal curve. If ''m'' < ''n'' then the ration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
