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Topic: Zariski tangent space

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Zariski topology

Oscar Zariski

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	Wikinfo \| Tangent space (Site not responding. Last check: 2007-10-21)
		All the tangent spaces have the same dimension, equal to the dimension of the manifold.
		For example, if the given manifold is a 2-sphere, one can picture the tangent space at a point as the plane which touches the sphere at that point and is perpendicular to the sphere's radius through the point.
		Once tangent spaces have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving on a manifold.
	www.internet-encyclopedia.org /wiki.php?title=Tangent_space (1222 words)

	Oscar Zariski Summary
		Oscar Zariski was one of the most influential mathematicians working in the field of algebraic geometry in the twentieth century.
		The Zariski topology, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions.
		Zariski became professor at Harvard University in 1947, retiring in 1969.
	www.bookrags.com /Oscar_Zariski (685 words)

	Britain.tv Wikipedia - Oscar Zariski
		Oscar Zariski (24 April 1899 - 4 July 1986) was a Belarusian-American mathematician, one of the most influential theorists of algebraic geometry in the 20th century.
		The Zariski topology, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions.
		Some of his major results, Zariski's main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck that ultimately unified algebraic geometry.
	www.britain.tv /wikipedia.php?title=Oscar_Zariski (629 words)

	Science Fair Projects - Tangent space
		The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.
		All the tangent spaces have the same dimension, equal to the dimension of the manifold.
		Once tangent spaces have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving on a manifold.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Tangent_vector (1370 words)

	Singularity theory at AllExperts
		Such singularities in algebraic geometry are the easiest in principle to study, since they are defined by polynomial equations and therefore in terms of a coordinate system.
		This result is often implicitly used to extend affine geometry to projective geometry: it is entirely typical for an affine variety to acquire singular points on the hyperplane at infinity, when its closure in projective space is taken.
		Technically this involves group actions of Lie groups on spaces of jets; in less abstract terms Taylor series are examined up to change of variable, pinning down singularities with enough derivatives.
	en.allexperts.com /e/s/si/singularity_theory.htm (1149 words)

	Tangent space - Wikipedia, the free encyclopedia
		The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.
		This defines an equivalence relation on such curves, and the equivalence classes are known as the tangent vectors of M at p.
		The relation between the tangent vectors defined earlier and derivations is as follows: if γ is a curve with tangent vector γ'(0), then the corresponding derivation is D(g) = (g o γ)'(0) (where the derivative is taken in the ordinary sense, since g o γ is a function from (-1,1) to R).
	en.wikipedia.org /wiki/Tangent_space (1221 words)

	tangent vectors Text - Physics Forums Library
		In summary, you could let the tangent vector v be the set of pairs of coordinate charts with v's coordinate representation, instead of letting v be the set of curves to which its tangent.
		So there exists a chart x such that the tangent of a and the tangent of b are equal under x and there exists a chart y such that the tangents of b and c are equal under y.
		This jacobian is an isomorphism (because f is diffeo) from the tangent space at x(p) to the tangent space at y(p), and...I want to say that the Jacobian takes tangents of curves to tangents of image curves, but that fact may rely on what I'm trying to prove.
	www.physicsforums.com /archive/index.php/t-94833.html (2268 words)

	Ringed space at AllExperts
		In mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space.
		Ringed spaces appear throughout analysis and are also used to define the schemes of algebraic geometry.
		is defined as the dual of this vector space.
	en.allexperts.com /e/r/ri/ringed_space.htm (855 words)

	Title
		The aim of the course is to explain an equivalent of such a result in the case of singular spaces, using intersection cohomology instead of classical cohomology theory.
		Some related problems will be discussed (bifurcation diagrams of quadratic differentials, spaces of defects in crystals, bifurcation diagrams of Smale functions) as well as the sightings of Stokes polyhedra and their relatives in other parts of mathematics.
		The tangent space to a leaf at a point $y$ is spanned by the values at $y$ of the liftable vector fields; its codimension equals the sum of the Tjurina numbers of $F$ at the singular points on the fibre over $y$.
	www.math.ruu.nl /people/siersma/asi-abstracts.html (2335 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Zariski Tangent Space - Example of cubics - Example of Special Linear Group - Theorem.
		An affine variety has a non empty open subset of points where the tangent space has minimal dimension.
		- Definition Smooth & Singular points - Intrinsic Definition of Tangent Space; The Map d_x, restriction to m_x, Theorem d_x : m_x/m_x^2 iso to dual of tangent space - Functoriality of tangent Space - Isomorphic = isomorphic tangent spaces - Algebraic Groups are smooth - Definition of dimension.
	www.math.tamu.edu /~frank.sottile/teaching/00A/697notes/Day11.txt (109 words)

	Reference.com/Encyclopedia/Tangent space
		The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.
		This defines an equivalence relation on such curves, and the equivalence classes are known as the tangent vectors of M at p.
		Specifically, if v is a tangent vector of M at a point p (thought of as a derivation) then define the directional derivative in the direction v by
	www.reference.com /browse/wiki/Tangent_space (1260 words)

	Oscar Zariski info here at en.my-widgets.com (Site not responding. Last check: 2007-10-21)
		That is Oscar Zariski the foremost point: the good of all, the good of the community, the good of humanity.
		As a healing principle, it means that, in the Oscar Zariski you are experiencing a block, there is someone Oscar Zariski you who needs you, just as you are.
		Another biography of Oscar Zariski appeared in the Gazette of the Australian Mathematical Society in the period 1988-1991 authorized by his wife of 60 years Yole Zariski.
	en.my-widgets.com /Oscar_Zariski (960 words)

	Deformation theory Information
		One expects, intuitively, that deformation theory, of the first order, should equate to the Zariski tangent space to a moduli space.
		where Θ is (the sheaf of germs of sections of) the holomorphic tangent bundle.
		The dimension of the moduli space, called Teichmüller space in this case, is computed as 3g − 3, by the Riemann-Roch theorem.
	www.bookrags.com /wiki/Deformation_theory (553 words)

	[No title]
		at the origin is the subspace to the tangent space to
		is isomorphic to the null space of the Jacobian.
		Even though the tangent space is the same as that of the previous case, we want to think of these as different kinds of singularities; we need better invariants to do so.
	odin.mdacc.tmc.edu /~krc/agathos/local.html (851 words)

	8.6 Vector Bundles and regular schemes
		Most of the examples of vector space schemes that we have seen so far are vector bundles; these are vector space schemes that are ``locally'' isomorphic to
		Note that any vector bundle is a vector space scheme and an exact sequence of vector bundles is also an exact sequence of vector space schemes.
		As a particular case we have the ``Jacobian criterion'' which says that a scheme is regular if the Zariski tangent vector space scheme is a vector bundle; note however that this is not in general necessary.
	www.imsc.res.in /~kapil/crypto/notes/node44.html (443 words)

	Logic Colloquium 2004, Tutorial on Geometric Stability Theory, Tangency (Site not responding. Last check: 2007-10-21)
		A smooth manifold M comes equipped with a tangent space at each point on the manifold.
		If M is embedded in Euclidean space, then the tangent space to M at a point x ∈ M may be identified with (a translate of) the affine subspace of the ambient Euclidean space best approximating M at x.
		For Zariski geometries, the notion of the tangent space itself is not available, but one can make sense of tangency.
	math.berkeley.edu /~scanlon/papers/LC04/LC04/tangent.html (268 words)

	Amazon.ca: The Geometry of Schemes: Books: David Eisenbud,Joe Harris (Site not responding. Last check: 2007-10-21)
		Projective schemes are the subject of Chapter 3, and are defined in terms of graded algebras and invariants of projective schemes embedded in projective space are discussed.
		The first one discussed is the notion of a flex, which deals (classically) with the locus of tangent lines to a variety.
		The flexes are defined in terms of the Hessian of the variety, the latter being generalized by the authors to define a scheme of flexes.
	www.amazon.ca /Geometry-Schemes-David-Eisenbud/dp/0387986383 (1319 words)

	Math 521: Introduction to Algebraic Geometry*, Fall 99
		Counting tangent lines to a fixed conic from a fixed point.
		Conics simultaneously containing 5 points (p.12) or tangent to 5 lines.
		Friday, Sept.3: There was a siign error in deriving the equation for the set of conics tangent to the line y=0.
	math.rice.edu /~hardt/521 (737 words)

	Price Compare ISBN 0387986375 The Geometry of Schemes by David Eisenbud - Direct Textbooks (Site not responding. Last check: 2007-10-21)
		The first one discussed is the notion of a flex, which deals (classically) with the locus of tangent lines to a variety.
		The flexes are defined in terms of the Hessian of the variety, the latter being generalized by the authors to define a scheme of flexes.
		They illustrate this for the Zariski tangent space.
	www.directtextbook.com /prices/0387986375 (1052 words)

	Zariski tangent space - Wikipedia, the free encyclopedia
		(The question of the origin comes up, when we take P as a general point on C; it is better to say 'affine space' and then note that P is a natural origin, rather than insist directly that it is a vector space.)
		which (thinking about affine schemes) allows one to speak in geometric terms, talking about tangent vectors.
		V.I. Danilov, "Zariski tangent space to an algebraic variety or scheme X at a point x" SpringerLink Encyclopaedia of Mathematics (2001)
	en.wikipedia.org /wiki/Zariski_tangent_space (467 words)

	Springer Online Reference Works
		, the Zariski tangent space is dual to the space
		Its sheaf of sections is called the tangent sheaf to
		The Zariski tangent space was introduced by O.
	eom.springer.de /Z/z099120.htm (137 words)

	[No title]
		We study the Zariski tangent space to the moduli space of integrable algebraic foliations, mainly of codimension one, in a complex projective space.
		Applying this to the problem of determining the irreducible components of the moduli space, we obtain alternative proofs of some known theorems and exhibit some new irreducible components, consisting of foliations with split tangent sheaf, which includes examples of foliations induced by group actions.
		A positive answer for this question was previously known in the case of plane curves, for homogeneous surfaces and for quasi homogeneous complex hypersurfaces with isolated singularity, as shown by Greuel in Constant Milnor number implies constant multiplicity for quasihomogeneous singularities, Manuscripta Math., 1986.
	w3.impa.br /~alga/alga05/abstracts.html (2475 words)

	Topics in geometry, fall 2006
		Class: irreducible spaces, decomposition of algebraic sets into irreducible components, irreducible algebraic sets correspond to prime ideals, affine varieties, coordinate rings, examples, quasi-affine varieties, noetherian rings, proof of the Hilbert Basissatz.
		Class: integrality in rings, Noether's normalisation lemma, proof of the Weak Nulstellensatz, projective n-space, homogeneous coordinates, affine coordinate charts, homogeneous polynomials, homogeneous ideals, algebraic sets in projective space, the Zariski topology, on the projective line we get the cofinite topology, the affine charts are open.
		Class: dimension, dimension becomes strictly smaller on a proper closed subvariety, Krull's Hauptidealsatz, topological characterisation of dimension, the tangent space, intuitive definition in affine space using linear parts, intrinsic definition (Zariski tangent space), this gives back the intuitive definition in affine space, singularity and non-singularity, Jacobi criterion.
	www.math.leidenuniv.nl /~rdejong/topics2006 (965 words)

	Toric varieties
		Zariski tangent space, smoothness and Jacobian criterion for affine varieties.
		Weighted projective space and fake weighted projective space.
		Sections of line bundles, linear systems and maps into projective space.
	www.mimuw.edu.pl /~jarekw/toric (461 words)

	HBA_Lectures on Curves on an Algebraic Surface
		The book also contains the infinitesimal study of the Picard scheme; i.e., the determination of the Zariski tangent space, not only for the Picard scheme but also for the subtle case of the reduced Picard scheme (in positive characteristic).
		The infinitesimal study of a universal curve is also given and this is related to what was called the completeness of the characteristic linear system of a good system of curves.
		Appendix: Re Representable Functors and Zariski Tangent Spaces
	www.hindbook.com /Home.asp?P=121 (288 words)

	Infinite dimensional algebraic geometry: algebraic structures on {$p$}-adic groups and their homogeneous spaces, ...
		In the first part of this paper we construct an algebraic theory of ind-schemes that allows us to represent finite $K$ schemes as infinite dimensional $k$-schemes and we apply this to semisimple groups.
		In the second part we construct spaces of lattices of fixed discriminant in the vector space $K^n.$ We determine the structure of these schemes.
		We devote particular attention to lattices of fixed discriminant in the lattice, $p^{-r}\mathcal O^n,$ computing the Zariski tangent space to a lattice in this scheme and determining the singular points.
	projecteuclid.org /getRecord?id=euclid.tmj/1113234835 (234 words)

	php-deluxe.net - description Zariski tangent space
		(The question of the origin comes up, when we take P as a general point on C ; it is better to say affine space and then note that P is a natural origin, rather than insist directly that it is a vector space.)
		The definition of singular point is then that the dimension of the tangent space is the dimension of an algebraic variety of V.
		For R coming from geometry over a field (mathematics) K, this will be a vector space over K.
	www.php-deluxe.net /wiwimod,index.page,Zariski-tangent-space.htm (451 words)

	Springer Online Reference Works
		, can be given the structure of a differentiable vector bundle, the tangent bundle.
		, one recovers the intuitive picture of the tangent space
		This point of view is more generally applicable and serves as the definition of tangent space in analytic and algebraic geometry, cf.
	eom.springer.de /t/t092200.htm (246 words)

	Citebase - Combinatorial Tangent Space and Rational Smoothness of Schubert Varieties
		Combinatorial Tangent Space and Rational Smoothness of Schubert Varieties
		We prove that the rational smoothness of a Schubert variety can be expressed in terms of a subspace of the Zariski tangent space called, the combinatorial tangent space.
		For this, we use a characterization of rational smoothness of a Schubert variety introduced by Carrell and Peterson [CP].
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0103022 (145 words)

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