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# Topic: Algebraic varieties

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 Kids.net.au - Encyclopedia Algebraic geometry - Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. In algebraic geometry, the geometric objects studied are defined as the set of zeroes of a number of polynomials: meaning the set of common zeroes, or equally the set defined by one or several simultaneous polynomial equations. Commutative algebra (as the study of commutative rings and their ideals) was developed by David Hilbert, Emmy Noether and others, also in the 20-th century, with the geometric applications in mind. www.kids.net.au /encyclopedia-wiki/al/Algebraic_geometry   (732 words)

 PlanetMath: algebraic geometry   (Site not responding. Last check: 2007-11-07) Algebraic geometry is the study of algebraic objects using geometrical tools. Algebraic groups are essentially matrix group schemes, and as such allow the tools of algebraic geometry to be applied to their study. Not every abelian variety is a Jacobian, but, for example, if one can find a curve lying on an abelian variety, there is a canonical homomorphism from the Jacobian of that curve to the abelian variety. planetmath.org /encyclopedia/AlgebraicGeometry.html   (2523 words)

 [No title]   (Site not responding. Last check: 2007-11-07) In algebraic geometry, the algebraic variety was the classical object of study. 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. www.informationgenius.com /encyclopedia/a/al/algebraic_variety.html   (359 words)

 Algebraic Geometry   (Site not responding. Last check: 2007-11-07) Algebraically, generic values of a parameter can be simulated by introducing a new transcendental element to the field of constants. Algebraic Geometry is intimately connected with projective geometry, commutative algebra, complex analysis, semi-algebraic geometry, singularity theory, topology, and number theory. Algebraic geometry does usually not deal with problems involving transcendental functions (sin, log, exp; these are dealt with in analytic geometry) and order relations (<,>: these are dealt with in semi-algebraic geometry). www.risc.uni-linz.ac.at /research/alggeo   (368 words)

 Algebraic geometry   (Site not responding. Last check: 2007-11-07) In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. The category of affine varieties is the dual category to the category of finitely generated reduced k-algebras and their homomorphisms. Varieties are subsumed in Alexander Grothendieck's concept of a scheme. www.sciencedaily.com /encyclopedia/algebraic_geometry   (1726 words)

 Algebraic varieties   (Site not responding. Last check: 2007-11-07) An affine algebraic variety was an irreducible algebraic setin 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. Anabstract algebraic variety would be a particular kind of locally ringed space, namely such that every point has a neighbourhood, as ringed space, of typeSpec(R) (spectrum of a ring) with R the co-ordinate ring of anaffine algebraic variety of the kind discussed in the first paragraph. These varieties have been called 'varieties in the sense of Serre', since Serre 's foundational paper FAC on sheaf cohomology was writtenfor them. www.therfcc.org /algebraic-varieties-220460.html   (362 words)

 Learn more about Algebraic geometry in the online encyclopedia.   (Site not responding. Last check: 2007-11-07) 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. Algebraic geometry was developed largely by the Italian geometers in the early part of the 20th century. Commutative algebra (as the study of commutative rings and their ideals) was developed by David Hilbert, Emmy Noether and others, also in the 20h century, with the geometric applications in mind. www.onlineencyclopedia.org /a/al/algebraic_geometry.html   (803 words)

 14: Algebraic geometry Algebraic geometry combines the algebraic with the geometric for the benefit of both. Singular Package: Singular is a computer algebra system for singularity theory and algebraic geometry developed by G.-M. Greuel, G. Pfister, H. Note that many computations in algebraic geometry are really computations in polynomials rings, hence computational commutative algebra applies. www.math.niu.edu /~rusin/known-math/index/14-XX.html   (523 words)

 Research in Algebraic Geometry Algebraic varieties are the solution sets to systems of polynomial equations. Abelian varieties are the group objects in the category of projective varieties. An elliptic curve is a variety defined by the equation, y^2=x^3+ax+b=0. www.math.duke.edu /~schoen/researchalggeo.html   (712 words)

 Valery Alexeev - Families of algebraic varieties associated with cell decompositions we associate: 1) a family of projective algebraic varieties, and 2) a family of pairs of projective algebraic varieties together with Cartier divisors. The parameter spaces of families of the second type for various cell decompositions fit together into a moduli space, a proper algebraic variety itself. One application of this construction is a canonical geometric compactification of the moduli of principally polarized abelian varieties. www.cms.math.ca /Events/winter98/w98-abs/node22.e   (124 words)

 Algebraic varieties. - Mathematics - What's Been Published   (Site not responding. Last check: 2007-11-07) Algebraic geometry I : complex projective varieties / David Mumford. Varieties of lattices / Peter Jipsen, Henry Rose. Geometry of higher dimensional algebraic varieties / Yoichi Miyaoka, Thomas Peternell. www.pitbossannie.com /rps-qa-algebraic-varieties.html   (69 words)

 ipedia.com: Algebraic geometry Article   (Site not responding. Last check: 2007-11-07) Given a subset V of which we know is a variety, it would be nice to determine the set of polynomials which generates it. A regular function on an algebraic set V contained in is defined to be the restriction of a regular function on, in the sense we defined above. It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space. www.ipedia.com /algebraic_geometry.html   (1723 words)

 Algebraic Geometry Conferences Homotopy theory for algebraic varieties with applications to K-theory and quadratic forms, MSRI (Berkeley, CA), 12-16 May 1998. Midwest Algebraic Geometry Conference, The University of Notre Dame, 7-9 November 1997. AGE workshop on algebraic surfaces (Fernando Serrano, in memoriam), University of Lisboa, 10-14 September 1997. eprints.math.duke.edu /conferences/alg-geom.html   (334 words)

 Citations: A singly-exponential stratification scheme for real semialgebraic varieties and its applications - Chazelle, ...   (Site not responding. Last check: 2007-11-07) We claim that the total number of such vertices that lie on the boundary of the fixed oe is O(q 2 (n) for some constant q depending on the degree and shape of the surfaces. : Let ff be one of the (constant number of) algebraic arcs that form the boundary of oe, and let H be the vertical surface formed by the union of all vertical rays whose bottom endpoints lie on ff. ....maximum degree (although the latter degree can grow quite fast with d) In what follows, we ignore the algebraic complexity of the subcells of the vertical decomposition, and will be mainly interested in bounding their number as a function of n, the number of given surfaces. citeseer.ist.psu.edu /context/77005/0   (3594 words)

 Algebraic Number Theory Archive   (Site not responding. Last check: 2007-11-07) math.NT/0411291: 12 Nov 2004, On the Manin-Mumford conjecture for abelian varieties with a prime of supersingular reduction, by Tetsushi Ito. math.NT/0408069: 4 Aug 2004, The arithmetic of Prym varieties in genus 3, by Nils Bruin. ANT-0102: 23 Mar 1998, Finite arithmetic subgroups of GL_n, by Marcin Mazur. front.math.ucdavis.edu /ANT   (12251 words)

 L-functions from algebraic geometry   (Site not responding. Last check: 2007-11-07) `Arithmetic algebraic geometry' is the modern name for the age-old theory of diophantine equations, with an emphasis on the use of tools from algebraic geometry. It is hoped that they provide the key to a deeper understanding of arithmetic properties of algebraic varieties, but they are still largely shrouded in mystery. The aim of the workshop is to explore general arithmetic and analytic properties of L-functions that arise from algebraic varieties. www.math.leidenuniv.nl /~psh/wan.shtml   (395 words)

 Amazon.com: Books: Algebraic Geometry (Graduate Texts in Mathematics)   (Site not responding. Last check: 2007-11-07) Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. It was the study of algebraic functions of one variable that led to the introduction of Riemann surfaces, and the later to a consideration of algebraic functions of two variables. Algebraic geometry is a huge and central subject that one cannot afford to waste time with the basics. www.amazon.com /exec/obidos/tg/detail/-/0387902449?v=glance   (2608 words)

 Arithmetic Algebraic Geometry Algebraic equations and their arithmetical properties have interested mankind since antiquity. The "Arithmetic Algebraic Geometry" network is sponsored by the European Commission's current programme for "Research Training Networks". In this framework, it is offering post- and pre-doctoral positions at each of its associated nodes, as part of its initiative to foster research in central subjects of this active domain of mathematical research. www.arithgeom-network.univ-rennes1.fr   (476 words)

 Amazon.com: Books: Algebraic Varieties (London Mathematical Society Lecture Note Series)   (Site not responding. Last check: 2007-11-07) Algebraic Geometry and Arithmetic Curves by Qing Liu An introduction to the theory of algebraic functions on varieties from a sheaf theoretic standpoint. For instance, an affine variety is defined as a space Y with a sheaf of k-algebras such that Mor(X,Y) is in natural bijection with k-Alg-Hom(k[Y],k[X]) for all other such spaces X which have a sheaf of k-algebras. www.amazon.com /exec/obidos/tg/detail/-/0521426138?v=glance   (1147 words)

 Number Theory Group, Oxford   (Site not responding. Last check: 2007-11-07) The project aims to investigate the density of rational points on algebraic varieties from the perspective of an analytic number theorist. Indeed the modern field of Arithmetic Geometry is ultimately concerned with the equivalent issue of the distribution of integral and rational points on algebraic varieties. For many algebraic varieties we expect there to be infinitely many integral points, and for many others we expect, but cannot prove, that there should be only finitely many such points. www.maths.ox.ac.uk /ntg/epsrc/epsrc.shtml   (450 words)

 Algebraic and finite groups of Lie type   (Site not responding. Last check: 2007-11-07) In the study of actions of algebraic groups on algebraic varieties, questions often arise concerning the fixed point spaces of elements or subgroups of the algebraic group in question. All simple algebraic groups are generated by a collection of so-called fundamental subgroups, which are subgroups SL(2) defined in terms of the root system of the algebraic group in question. Many papers have been written on the structure of subgroups which contain certain elements of these fundamental subgroups; probably the most studied are the root elements, which are unipotent elements in such SL(2) subgroups. www.ma.ic.ac.uk /~mwl/alg.htm   (236 words)

 Affine and Projective varieties   (Site not responding. Last check: 2007-11-07) Prolongations of infinitesimal linear automorphisms of projective varieties and... 1.3.2 Algebraic Geometry -- Dr Mavlyutov -- 16 HT... 4 Zero cycles on varieties of higher dimension... www.scienceoxygen.com /math/736.html   (61 words)

 Partial Zeta Functions of Algebraic Varieties over Finite Fields - Wan (ResearchIndex)   (Site not responding. Last check: 2007-11-07) Motivated by arithmetic applications, we introduce the notion of a partial zeta function which generalizes the classical zeta function of an algebraic variety defined over a finite field. The first approach, using an inductive fibred variety point of view, shows that the partial zeta function is rational in an interesting case, generalizing Dwork's... 2 functions from algebraic geometry over finite fields (context) - Wan, L- the rationality of the zeta function of an algebraic varieti.. citeseer.ist.psu.edu /wan01partial.html   (509 words)

 Schubert Varieties Flag varieties are projective algebraic varieties, homogeneous under an action of a linear algebraic group. Their closures are the Schubert varieties; these may be singular, but admit nice desingularizations. The Grothendieck ring of vector bundles on an algebraic variety provides a "generalization" of the usual cohomology or Chow rings, which gives rise to a more refined intersection theory. www.mimuw.edu.pl /~aweber/schubert   (665 words)

 algebraic varieties   (Site not responding. Last check: 2007-11-07) We may use Spec to create an affine scheme (or algebraic variety) with a specified coordinate ring and ring to recover the ring. We may use Proj to create a projective scheme (or algebraic variety) with a specified homogeneous coordinate ring. The most important reason for introducing the notion of algebraic variety into a computer algebra system is to support the notion of coherent sheaf. www.mi.uni-koeln.de /b/Macaulay2/html/741.html   (65 words)

 AMCA: On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex ...   (Site not responding. Last check: 2007-11-07) We prove that for any affine variety S defined over Q there exist an Artin group G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety Hom(G, PO(3))//PO(3). Suppose M is a smooth connected complex algebraic variety with the fundamental group G, L is a reductive algebraic Lie group and f: G --> L is a representation with finite image. The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. at.yorku.ca /cgi-bin/amca/cadi-14   (253 words)

 LMS/EPSRC Short Course   (Site not responding. Last check: 2007-11-07) Algebraic geometry occupies a central place in modern pure mathematics, with connections to number theory, theoretical physics and differential geometry in particular. For example, elliptic curves and modular curves play vital roles in arithmetic; startling advances in the theory of higher-dimensional varieties and moduli spaces have emerged from, and contributed to, physics; and the theory of real 4-manifolds has similarly interacted with complex algebraic surfaces. In part because of its many connections, algebraic geometry is often seen as being hard to learn, and is left in the hands of specialists. www.bath.ac.uk /~masgks/ShortCourse   (278 words)

 Publications of the SPACES team Five algebraic provers based on Wu's method of characteristic sets, the GrÃ¶bner basis method, and other triangularization techniques are available for proving such theorems in elementary (and differential) geometry. The use of perfomant algebra tools applied to geometry in a computer algebra environment opens the way to all exact solutions to the kinematics problem by the implementation of perfomant tools such as algebraic system representation, Groebner bases, rational univariate representation RUR and real root isolation. In particular, an algebraic relation between the first and the second fundamental coefficients in a very compact form has been derived, which is more general and has smaller degree than a relation discovered previously by Z. Li. www-calfor.lip6.fr /~safey/Spaces/publications.html   (13078 words)

 Exercises In Algebraic Varieties Everyone else in my class is confused about the exercises in algebraic varieties and the teacher just isn't explaining it like she should. The Algebra Helper software will literally help you work on your own algebra problems at your own pace. I need help with exercises in algebraic varieties and the math labs are no help at all and my friends are no help. www.algebra-answer.com /algebra-helper/exercises-in-algebraic-varieties.html   (470 words)

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