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Topic: Affine varieties

	Algebraic variety - Wikipedia, the free encyclopedia
		An affine algebraic variety is essentially the set of common zeroes of a set of polynomials, and is one of the central objects of study in classical (and to some extent, modern) algebraic geometry.
		An abstract algebraic variety is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings.
		Basically, a variety is a scheme whose structure sheaf is a sheaf of K-algebras with the property that the rings R that occur above are all domains and are all finitely generated K-algebras, i.e., quotients of polynomial algebras by prime ideals.
	en.wikipedia.org /wiki/Projective_varieties (955 words)

	Algebraic geometry - Wikipedia, the free encyclopedia
		The category of affine varieties is the opposite category to the category of finitely generated reduced k-algebras and their homomorphisms.
		Varieties are subsumed in Alexander Grothendieck's concept of a scheme.
		An important class of varieties, not easily understood directly from their defining equations, are the abelian varieties, which are the projective varieties whose points form an abelian group.
	en.wikipedia.org /wiki/Algebraic_geometry (1802 words)

	Algebraic group - Wikipedia, the free encyclopedia
		In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety.
		According to another basic theorem, any group in the category of affine varieties has a faithful linear representation: we can consider it to be a matrix group over K, defined by polynomials over K and with group operation simply matrix multiplication.
		Affine group scheme is the concept dual to a type of Hopf algebra.
	en.wikipedia.org /wiki/Algebraic_group (380 words)

	PlanetMath: variety
		A projective variety is identified as the gluing together of the affine varieties obtained by taking the complements of hyperplanes.
		Using the theory of schemes, we can glue these affine varieties together to get a scheme; the result will be projective.
		This is version 5 of variety, born on 2004-03-26, modified 2004-04-06.
	planetmath.org /encyclopedia/Variety.html (270 words)

	No Title
		We compute the asymptotics of the number of integer points on affine homogeneous varieties G/H, under the assumption that H is an affine symmetric subgroup of G. We use the Howe-Moore theorem and a certain geometric property of affine symmetric spaces.
		We study the density of integer points on affine homogeneos varieties G/H, such as the set of matrices with a given characteristic polynomial.
		We study a refined version of the Linnik problem on the asymptotic behavior of the number of representations of integer $m$ by an integral polynomial as $m$ tends to infinity.
	www.math.uchicago.edu /~eskin/abstracts.html (1599 words)

	PlanetMath: affine variety
		The definition above has the extra condition that an affine variety not be the union of two closed subsets, i.e.
		A quasi-affine variety is then an open set (in the Zariski topology) of an affine variety.
		This is version 7 of affine variety, born on 2001-12-21, modified 2005-05-06.
	planetmath.org /encyclopedia/AffineVariety.html (275 words)

	Algebraic variety -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		An affine algebraic variety is defined to be an irreducible algebraic set in some (Click link for more info and facts about affine space) affine space, over an (Click link for more info and facts about algebraically closed field) algebraically closed field K.
		An abstract algebraic variety is a particular kind of (An elaborate and systematic plan of action) scheme.
		These varieties have been called 'varieties in the sense of Serre', since (Click link for more info and facts about Serre) Serre's foundational paper FAC on (A package of several things tied together for carrying or storing) sheaf cohomology was written for them.
	www.absoluteastronomy.com /encyclopedia/a/al/algebraic_variety.htm (851 words)

	Algebraic geometry (Site not responding. Last check: 2007-11-05)
		The category of affine varieties is the dual category to the category of finitely generated reduced k-algebras and their homomorphisms.
		An important class of varieties are the abelian varieties which are varieties whose points form an abelian group.
		The prototypical examples are the elliptic curves that were instrumental in the proof of Fermat's last theorem and are also used in elliptic curve cryptography.
	www.sciencedaily.com /encyclopedia/algebraic_geometry (1736 words)

	Algebraic geometry Article, Algebraicgeometry Information (Site not responding. Last check: 2007-11-05)
		It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is verysimilar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extendsto the ambient topological space.
		The category of affine varieties is the dualcategory to the category of finitely generated reduced k- algebras and theirhomomorphisms.
		Varieties are subsumed in Alexander Grothendieck 's concept of a scheme.
	www.anoca.org /set/regular/algebraic_geometry.html (1668 words)

	DC MetaData for:Higher K-theory of toric varieties
		A natural higher K-theoretic analogue of the triviality of vector bundles on affine toric varieties is the conjecture on nilpotence of the multiplicative action of the natural numbers on the K-theory of these varieties.
		Moreover, it yields a similar behavior of not necessarily affine toric varieties and, further, of their equivariant closed subsets.
		The conjecture is equivalent to the claim that the relevant admissible morphisms of the category of vector bundles on an affine toric variety can be supported by monomials not in a non-degenerate corner subcone of the underlying polyhedral cone.
	www.mathematik.uni-osnabrueck.de /preprints/shadow/calg0104.rdf.html (195 words)

	Varieties, Ideals, Nullstellensatz (Site not responding. Last check: 2007-11-05)
		Projective varieties can be thought of as ``completions'', ``compactifications'', or ``closures'' of affine varieties.
		Conversely, affine varieties can be thought of as building blocks of projective varieties (indeed, they constitute an open cover), and hence local properties are easier to describe using affine varieties.
		However, projective varieties vary ``nicely'' in families and hence parametrizing and moduli spaces are usually constructed for projective varieties with certain defining common properties.
	mathcircle.berkeley.edu /BMC3/alg-geom/node1.html (465 words)

	Glossary
		An algebraic map or regular map or morphism of quasiprojective varieties is a map of whose graph is closed.
		The functor X → A(X) is an antiequivalence of the full subcategory of affine varieties and the category of affine domains over k (finitely generated k-algebras which are integral domains).
		A homogenous space is a variety X such that there is an algebraic group G and a transitive action on X for which GxX → X is a morphism.
	www.math.purdue.edu /~dvb/algeom2.html (786 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Y, where X and Y are affine varieties, both nonempty and distinct from A
		Let X be an affine variety (respectively a projective variety).
		Then X is the union of finitely many irreducible affine varieties (respectively finitely many irreducible projective varieties).
	www.math.umn.edu /~roberts/math8203/irred_var.html (839 words)

	AMCA: Ring-theoretic Properties of Affine Semigroups by Ngo Viet Trung (Site not responding. Last check: 2007-11-05)
		Affine semigroups are finitely generated submonoids of the additive monoids N
		The name `affine semigroups' comes from the facts that their semigroup rings are coordinate rings of affine toric varieties.
		The aim of the talk is to show how important algebraic properties of affine toric varieties can be described by means of numerical and combinatorical data of the underlying affine semigroups.
	at.yorku.ca /c/a/e/c/47.htm (118 words)

	Fields Institute - Workshop on Shimura varieties and related topics (Site not responding. Last check: 2007-11-05)
		We try to prove that any scheme automorphism of each geometrically irreducible component of a certain Shimura variety of PEL type modulo $p$ (of prime-to-$p$ level) is given by an isogeny action (generalizing a result of Shimura in characteristic 0).
		Over a non archimedian local field of equal characteristics, the orbital integrals which enter in the statement of the ``Fundamental Lemma'' for unitary groups are directly related to the affine Springer fibers for the general linear groups and it is not difficult to formulate a geometric conjecture which implies the ``Fundamental Lemma''.
		As a result of this analysis, weobtain a description of the l-adic cohomology of the Shimura varieties in terms of the l-adic cohomology with compact supports of the Igusa varieties and of the Rapoport-Zink spaces, in the appropriate Grothendieck group.
	www.fields.utoronto.ca /programs/scientific/02-03/automorphic_forms/shimura/abstracts.html (1764 words)

	Publications
		The first is a mixed version of the equivariant derived category of X and the second is a mixed version of the derived category of sheaves on Y which are locally constant with unipotent monodromy on each orbit.
		This result is then used to explain the examples of reducible characteristic varieties of Schubert varieties given by Kashiwara and Saito in the full flag variety for type A and by Boe and Fu for the Lagrangian Grassmannian.
		This nonnegativity comes from showing that the restriction of the intersection cohomology sheaf on a toric variety to the closure of an orbit is a direct sum of intersection homology sheaves.
	www.math.umass.edu /~braden/papers.html (764 words)

	ipedia.com: Algebraic geometry Article (Site not responding. Last check: 2007-11-05)
		Given a subset V of which we know is a variety, it would be nice to determine the set of polynomials which generates it.
		Using regular functions from an affine variety to, we can define regular functions from one affine variety to another.
		If V' is a variety contained in, we say that f is a regular function from V to V' if the range of f is contained in V'.
	www.ipedia.com /algebraic_geometry.html (1723 words)

	Alistair Savage - Research Interests
		In particular, we have a quiver variety for each weight space and the number of irreducible components of this variety is equal to the dimension of the weight space.
		The quiver varieties of Nakajima are an extension to more general type of the varieties of affine type which were introduced by Kronheimer and Nakajima in their description of Yang-Mills instantons on gravitational instantons.
		One of the important results from the theory of quiver varieties is the definition of the canonical and semicanonical bases in universal enveloping algebras and their representations which have remarkable properties.
	www.math.toronto.edu /alistair/research.html (609 words)

	[No title]
		\section {Construction of toric varieties} \subsection {Finitely generated algebras and affine varieties} The prerequisite of this course is some basic knowledge of elementary commutative algebra, algebraic geometry, convex bodies, and algebraic topology.
		By definition, an affine variety is the common solution set of some polynomials.
		For an affine variety, the regular function ring is finitely generated algebra.
	www.math.sunysb.edu /~bzhang/toric-note.txt (1065 words)

	Algebraic Curves
		The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal in a polynomial ring in several variables.
		There is a close relationsship between ideals and varieties which reveals the intimate link between algebra and geometry.
		Describe, implement and apply an algorithm for determining the smallest projective variety containing a given affine variety.
	home.imf.au.dk /matjph/kurver03.html (595 words)

	geomsem05
		But it is known that in general the geometric infinitesimal variation of Hodge structures are "lost" among the crowd of those integral elements.
		Schubert varieties are Gorenstein, analogous to Lakshmibai and
		These are real varieties X which have a nice cellular decomposition after base extension to the complex numbers.
	www.math.tamu.edu /~jml/geomsem05.html (1113 words)

	1.3.2 Algebraic Geometry -- Dr Mavlyutov -- 16 HT (Site not responding. Last check: 2007-11-05)
		Algebraic geometry is the study of algebraic varieties: an algebraic variety is, roughly speaking, a locus defined by polynomial equations.
		It is geometry based on algebra rather than on calculus, but over the real or complex numbers it provides a rich source of examples and inspiration to other areas of geometry.
		Affine algebraic varieties, the Zariski topology, morphisms of affine varieties, coordinate rings, regular functions, irreducible varieties.
	www.maths.ox.ac.uk /current-students/undergraduates/handbooks-synopses/2004/html/sect-c-04/node13.html (152 words)

	Algebraic Geometry
		A projective variety is a consideration how to fit the algebraic variety and affine variety together, to make an algebraic variety in projective space.
		An affine variety is a k-space which is isomorphic to a closed subspace of an affine space over k.
		A variety (in the sense of Serre) is a k-prevariety which is separated and has a finite affine open cover.
	www.risberg.ws /Hypertextbooks/Mathematics/Geometry/algebraic.htm (1388 words)

	Introduction to projective varieties (Site not responding. Last check: 2007-11-05)
		My approach consists of avoiding all the algebraic preliminaries that a standard algebraic geometry course uses for affine varieties and thus start directly with projective varieties (which are the varieties that have good properties).
		The main technique I use is the Hilbert polynomial, from which it is possible to rigorously and intuitively introduce all the invariants of a projective variety (dimension, degree and arithmetic genus).
		The price to pay for this shortcut is that the way to produce the important results (the most important one for practical purposes is the theorem about the dimension of the fibers) is not always clear, since many results or even definitions have local nature.
	www.mat.ucm.es /~arrondo/projvar.html (423 words)

	Math 273 (Site not responding. Last check: 2007-11-05)
		The image of a projective variety under a morphism, the image of a quasi-projective variety, how the dimensions of fibers behave and why this is significant, families of varieties, incidence correspondences and their uses, parametrizability and non-parametrizability.
		Become familiar with specific geometric objects which are important in many areas of pure mathematics: Grassmannians, the simplest flag varieties, Segre, and Veronese varieties, quadric hypersurfaces.
		Introduce the local and infinitesimal study of algebraic varieties: Tangent space, singular locus, closedness of singular locus, tangent variety.
	www.math.duke.edu /~schoen/alg27398.html (355 words)

	Affine Schubert Varieties (Site not responding. Last check: 2007-11-05)
		The variety of two-step loop-complexes is the set of pairs of matrices {(X,Y) Using the construction of Lusztig [4], we show that this variety is isomorphic to an open subset of a Schubert variety for the loop group of GL n
		The affine Grassmannian Gr(V) is the space of all A-lattices of V. Clearly Gr(V) is a homogeneous space with respect to the obvious action of G, and the stabilizer of the standard lattice E is the maximal parabolic P: = GL n
		We have the corresponding decomposition of the affine flag variety Fl(a,b;V) =
	www.mth.msu.edu /~magyar/AffineNotes.html (1181 words)

	Publication list of O. Kharlampovich
		O. Kharlampovich, The word problem for varieties of groups and Lie algebras; the boundary between solvability and unsolvability, in "Ann.
		O. Kharlampovich, Algorithmic problems for the varieties of groups and Lie algebras, in "Annals of the 10th All-Soviet Conference on Math.
		O. Kharlampovich, A. Myasnikov, Description of fully residually free groups and irreducible affine varieties over a free group, Summer School in group theory in Banff, CRM Proceedings and Lecture Series, 1998, 71--80.
	www.math.mcgill.ca /olga/complete.html (887 words)

	Algebraic Groups Research Seminar (Site not responding. Last check: 2007-11-05)
		We show that normal affine G-varieties that possess an open orbit isomorphic to G are completely determined by an associated set of one-parameter subgroups of G and that such sets are "strongly convex rational polyhedral cones" in a sense generalizing the definition in toric geometry.
		Abstract: We show that an affine embedding X of an algebraic group G is determined by the set of one-parameter subgroups of G which have a limit in X.
		I want to discuss the 1983 paper "Plongements d'espaces homogenes" by Luna and Vust in which the authors relate the classification of all such embeddings to the study of a space of local rings in k(G/H), which is given a Zariski topology.
	www.math.uiuc.edu /~dcmurphy/seminars/groups.html (1398 words)

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