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Topic: Projective varieties

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	Algebraic variety - Wikipedia, the free encyclopedia
		Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) algebraic geometry.
		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.
		Projective algebraic manifolds are an equivalent definition for projective varieties.
	en.wikipedia.org /wiki/Algebraic_variety (1098 words)

	Algebraic geometry - Wikipedia, the free encyclopedia
		While projective geometry was originally established on a synthetic foundation, the use of homogeneous coordinates allowed the introduction of algebraic techniques.
		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 (1720 words)

	PlanetMath: variety (Site not responding. Last check: 2007-11-01)
		a variety would appear to conflict with the preexisting notion of an affine or projective variety.
		A projective variety is identified as the gluing together of the affine varieties obtained by taking the complements of hyperplanes.
		This is version 5 of variety, born on 2004-03-26, modified 2004-04-06.
	www.planetmath.org /encyclopedia/variety.html (270 words)

	Algebraic variety (Site not responding. Last check: 2007-11-01)
		An affine algebraic variety was an irreducible algebraic set in some affine space, over an algebraically closed field K. It therefore was given by a co-ordinate ring that was an integral domain, a quotient of a polynomial ring over K by a prime ideal.
		An abstract algebraic variety would be a particular kind of locally ringed space, namely such that every point has a neighbourhood, as ringed space, of type Spec(R) (spectrum of a ring) with R the co-ordinate ring of an affine algebraic variety of the kind discussed in the first paragraph.
		This definition had the big advantage of allowing varieties which were complete, in the sense of algebraic geometry, but not given as projective varieties (which are complete, but now that became an intrinsic concept).
	www.serebella.com /encyclopedia/article-Algebraic_variety.html (1008 words)

	Varieties, Ideals, Nullstellensatz
		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)

	Algebraic geometry Article, Algebraicgeometry Information (Site not responding. Last check: 2007-11-01)
		The category of affine varieties is the dualcategory to the category of finitely generated reduced k- algebras and theirhomomorphisms.
		While projective geometry was originally established on a synthetic foundation, the use of homogenousco-ordinates allowed the introduction of algebraic techniques.
		Varieties are subsumed in Alexander Grothendieck 's concept of a scheme.
	www.anoca.org /set/regular/algebraic_geometry.html (1668 words)

	Prof. Brodmann
		COHOMOLOGY OF PROJECTIVE SCHEMES: This is the geometric counterpart of a particular branch of Local Cohomology Theory.
		Cohomology of Projective Schemes allows to assign to a pair (X,F), consisting of a projective scheme X (over a field for example) and a coherent sheaf F over X a string of numerical functions, the so called cohomological Hilbert functions of (X,F).
		PROJECTIVE VARIETIES OF LOW DEGREE: The degree deg(X) of a non-degenerate irreducible variety in projective r-space (over an algebraically closed base field K) satifies the inequality deg(X)>r-dim(X), where dim(X) denotes the dimension of X. Varieties of minimal degree, thus varieties with deg(X)=r-dim(X)+1 are rather well understood.
	www.math.unizh.ch /brodmann (881 words)

	Projective Varieties
		In contrast, projective space consists of lines passing through the origin, such as the x axis, or 2x = 5y = 3z.
		Thus the points of the sphere define projective space, as long as you remember that each point is essentially the same as the point on the other side.
		In summary, the polynomials that vanish on specific regions in projective space are the homogeneous polynomials, where each term has degree d.
	www.mathreference.com /ag-pv,intro.html (458 words)

	Science Fair Projects - Algebraic geometry
		First we will define a regular function from a variety into affine space: Let V be a variety contained in
		The category of affine varieties is the opposite category to the category of finitely generated reduced k-algebras and their homomorphisms.
		Science kits, science lessons, science toys, maths toys, hobby kits, science games and books - these are some of many products that can help give your kid an edge in their science fair projects, and develop a tremendous interest in the study of science.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Algebraic_equation (1927 words)

	Algebraic varieties. - Mathematics - What's Been Published (Site not responding. Last check: 2007-11-01)
		Algebraic geometry I : complex projective varieties / David Mumford.
		Tangents and secants of algebraic varieties / F.L. Zak.
		Geometry of higher dimensional algebraic varieties / Yoichi Miyaoka, Thomas Peternell.
	www.pitbossannie.com /rps-qa-algebraic-varieties.html (69 words)

	Grassmannian
		a topological space, homogeneous space, differential manifold or algebraic variety), and notice that up to appropriate isomorphisms, we have a well-defined geometric object for the given pair (n,k).
		Supposing first that K is the real number or complex number field, the easiest approach to Grassmannians is probably to consider them as homogeneous spaces.
		Explicit homogeneous coordinates, the Plücker coordinates, are known, and come from the k-th exterior power: apply the wedge product to a basis of a k-dimensional subspace and the resulting k-vector is well-defined, up to a scalar multiple.
	www.brainyencyclopedia.com /encyclopedia/g/gr/grassmannian.html (604 words)

	Glossary
		A quasiprojective variety is an open subset of a projective variety.
		An algebraic map or regular map or morphism of quasiprojective varieties is a map of whose graph is closed.
		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)

	Arkhipov Abstract (Site not responding. Last check: 2007-11-01)
		Given a smooth projective variety $X$, we define a certain ind-scheme $L(X)$ playing the role of "Laurent loops with values in $X$".
		We introduce the notion of semi-infinite cohomology of an ind-scheme and consider the semi-infinite cohomology with trivial coefficients of the projectivization of a Tate vector space $V$ and of the Grassmannian of k-dimensional subspaces in $V$.
		We conclude the talk with some conjectures connecting semi-infinite cohomology of Laurent loops with values in a smooth projective variety and quantum cohomology of this variety.
	darkwing.uoregon.edu /~dps/Colloquium/Abstracts/arkhipov.html (154 words)

	Computations in algebraic geometry with Macaulay 2
		The duals of these infinite projective resolutions are finitely generated differential graded modules over a graded polynomial ring, so they can be represented in the computer, and can be used to compute Ext modules simultaneously in all homological degrees.
		All components of the scheme are toric varieties, and among them, there is a fairly well understood coherent component.
		D-modules and Cohomology of Varieties, by Uli Walther:
	www.math.uiuc.edu /Macaulay2/Book (943 words)

	Linear Algebraic Groups and Related Structures (Site not responding. Last check: 2007-11-01)
		These are projective varieties, homogenous under the action of the group.
		It is well known that the first is a 5-dimensional projective quadric (associated to a Pfister neighbour).
		We will establish here that, even th two varieties are not isomorphic in the category of algebraic varieties, they become so in the category of correspondences.
	www.mathematik.uni-bielefeld.de /LAG/man/114.html (212 words)

	Varieties and Schemes for Dummies, Part III \| The String Coffee Table
		A projective algebraic variety is an irreducible algebraic subset in
		In a sense, projective varieties are like a first tiny step from affine varieties to schemes, since every projective variety can be covered by affine varieties.
		One upshot of all this is that we get an intrinsic definition of what a variety is, one that does not rely on constructing it as a set of common zeros of a collection of polynomials.
	golem.ph.utexas.edu /string/archives/000852.html (1616 words)

	Introduction to projective varieties (Site not responding. Last check: 2007-11-01)
		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)

	TEL :: CONSULTER
		We study the homogenous projective varieties $X(\alpha_1)$ and $X(\alpha_2)$ associated to each two roots of a group of type $G_2$.
		The first one, $X(\alpha_1)$, is a 5-dimensional projective quadric associated to a Pfister neighbour and the second one, $X(\alpha_2)$, is a Fano variety (of genus 10).
		They are not isomorphic as algebraic varieties but they become isomorphic as objects of the category of correspoondences (and of consequently as objects in the category of Chow motives).
	tel.ccsd.cnrs.fr /documents/archives0/00/00/42/14/index_fr.html (711 words)

	UIUC Dept. of Mathematics Seminar Calendar
		Abstract: After finishing our discussion on the open cover of a projective variety by affine varieties, we move on to the new topic of the projective closure of an affine variety.
		The common generalization of affine and projective varieties to quasi-projective varieties provides the framework to study these objects simulataneously as well as enlarging the class of varieties significantly.
		Then we'll relate properties of the hilbert polynomial of a projective variety to dimension and degree of the variety and discuss invariants.
	torus.math.uiuc.edu /cal/math/cal?year=2005&month=07&day=01&interval=next+12+months®exp=Algebraic+Geometry (3802 words)

	Projective Algebraic Set (Site not responding. Last check: 2007-11-01)
		Note that our projective space could be based on k, or an algebraic extension of k, or its algebraic closure, while the polynomials of s could have coefficients in k, or any subring of k.
		Conversely, a region r in projective space defines a set of homogeneous polynomials r′ that vanish on r.
		A homogeneous ideal is also an ideal, hence, using the same set of polynomials s, an algebraic set in projective space is an algebraic set in normal space.
	www.mathreference.com /ag-pv,algset.html (493 words)

	Ian Morrison: Selected Publications
		An evaluation of old and new approaches to verifying stability of Hilbert points of projective varieties with speculations on where more effective methods might be found.
		Relates the initial forms of the ideal of a projective variety to the geometric invariant theory of its Hilbert point(s).
		For a vector bundle E over a curve C, shows that bundle-stability of E and stability of suitable projective models of P(E) are equivalent.
	www.fordham.edu /morrison/publications.html (833 words)

	Geometry.Net - Pure And Applied Math Books: Convex Geometry
		This book is an advanced overview of the theory of toric varieties written for individuals with a strong background in algebraic geometry, topology, and algebra.
		The fundamental group of a toric variety is given an explicit characterization, but the proof is omitted unfortunately.
		The Hironaka resolution of singularities theorem is discussed for toric varieties, the proof being a lot simpler of course in this case.
	www.geometry.net /pure_and_applied_math_bk/convex_geometry.html (683 words)

	Euclid - server of EAGER (Site not responding. Last check: 2007-11-01)
		In algebraic-geometric terms, certain homogeneous stable vector bundles on an algebraic variety correspond after dimensional reduction, using the results of Garcia-Prado and Bradlow, to a stable pair on the quotient variety.
		The aim of this research plan is an investigation into the projective geometry of the moduli spaces of rank-2 vector bundles.
		It is a locally factorial, irreducible, projective variety.
	www.imar.ro /exchanges/euclid.html (1257 words)

	OUP: UK General Catalogue
		This book is an account of the combinatorics of projective spaces over a finite field, with special emphasis on one and two dimensions.
		With its successor volumes, Finite projective spaces over three dimensions (1985), which is devoted to three dimensions, and General Galois geometries (1991), on a general dimension, it provides a comprehensive treatise of this area of mathematics.
		The first part comprises two chapters, the first of which is a survey of finite fields; the second outlines the fundamental properties of projective spaces and their automorphisms, as well as properties of algebraic varieties and curves, in particular, that are used in the rest of the book and the accompanying two volumes.
	www.oup.co.uk /isbn/0-19-850295-8 (590 words)

	[No title]
		Chapter 3: Affine varieties, ps or pdf (Revised June 2005)
		Chapter 9: Projective Geometry, ps or pdf (Revised 2/17/04)
		Chapter 11: Projective elimination theory, ps or pdf
	www.math.rice.edu /~hassett/CAGbook/CAGtoc.html (151 words)

	K-Theory of non-linear projective toric varieties, by Thomas Huettemann (Site not responding. Last check: 2007-11-01)
		K-Theory of non-linear projective toric varieties, by Thomas Huettemann
		By analogy with algebraic geometry, we define a category of non-linear sheaves (quasi-coherent homotopy-sheaves of topological spaces) on projective toric varieties and prove a splitting result for its algebraic K-theory, generalising earlier results for projective spaces.
		The splitting is expressed in terms of the number of interior lattice points of dilations of a polytope associated to the variety.
	www.math.uiuc.edu /K-theory/0752 (108 words)

	Department Mathematics - Analytic and Algebraic geometry
		The main research areas are in the field of Algebraic Geometry, and there are some researches in the area of Complex Analysis.
		This consider the three main classes of projective varieties, namely curves, surfaces and higher dimensional varieties.
		In the case of surfaces the main objects of study are the moduli spaces of minimal surfaces of general type with given (small) value of invariants.
	www.unitn.it /dipartimenti/mate/research_group/analytic_algebraic_geometry.php (141 words)

	Visualization of some simple algebro-geometric ideas (Site not responding. Last check: 2007-11-01)
		Animations have been also used to demonstrate the notion of projective equivalence.
		This projection, naturally, deforms the surface and does not give its real picture.
		In this animation we see projective equivalence, or equivalence under the action of the group of collineations PGL(2,C), of two seemingly different plane algebraic third order curves.
	members.tripod.com /vismath2/lip/index.html (879 words)

	Award#0306487 - Uniformization of Projective Manifolds and Global Properties of Embeddable CR-manifolds
		The second topic of this project is about the interaction between the properties of special real submanifolds and the properties of the complex manifolds that contain them.
		A flexible construction of projective varieties with given prescribed singularities shows that even though CR-submanifolds impose restrictions in analytical properties of the ambient complex manifold like the Kodaira dimension, they are expected to be less restrictive on topological properties like the fundamental group.
		This project will also allow the principal investigator to organize seminars that will bring experts of the field from other parts of the country to his institution.
	www.fastlane.nsf.gov /servlet/showaward?award=0306487 (472 words)

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