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Topic: Singular point of an algebraic variety

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In the News (Thu 22 Aug 19)

  Singular point of an algebraic variety - Wikipedia, the free encyclopedia
In mathematics, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that V is not locally flat there.
A general algebraic variety V being defined by several polynomials, or in algebraic terms an ideal of polynomials, the condition on a point P to be a singular point of V is that none of those polynomials have a non-zero linear (degree 1) term, when written in terms of variables
Points of V that are not singular are non-singular.
en.wikipedia.org /wiki/Singular_point_of_an_algebraic_variety   (462 words)

 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.
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/Algebraic_variety   (1098 words)

 Algebraic Number Theory Archive   (Site not responding. Last check: 2007-11-01)
math.NT/0408069: 4 Aug 2004, The arithmetic of Prym varieties in genus 3, by Nils Bruin.
ANT-0342: 28 Mar 2002, On an Archimedean analogue of Tate's conjecture, by Dipendra Prasad and C.S.Rajan.
ANT-0185: 7 Jun 1999, An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod-p cohomology of GL(n,Z), by Avner Ash and Warren Sinnott.
front.math.ucdavis.edu /ANT   (12251 words)

 PlanetMath: nonsingular variety
Over the real or complex numbers, nonsingularity corresponds to “smoothness”: at nonsingular points, varieties are locally real or complex manifolds (this is simply the implicit function theorem).
Singular points generally have “corners” or self intersections.
This is version 5 of nonsingular variety, born on 2001-12-21, modified 2002-12-27.
planetmath.org /encyclopedia/NonsingularVariety.html   (125 words)

 Singular Tutorial - Elimination
Elimination is the algebraic counterpart of the geometric concept of projection.
This is achieved by an ordering with the block of t-variables having a global ordering (and coming before the x-variables) and the x-variables having a local ordering.
The tangent space at a smooth point F(t1,...,tr) // is given as the image of the tangent space at (t1,...,tr) under // the tangent map (affine coordinates) // T(t1,...,tr): (y1,...,yr) --> jacob(f)*transpose((y1,...,yr)) // where jacob(f) denotes the jacobian matrix of f with respect to the // t's evaluated at the point (t1,...,tr).
www.singular.uni-kl.de /Tutor-1.2/tutor_19.html   (692 words)

 [No title]
Algebraic geometry is one of the oldest and vastest branches of mathematics.
In McGill, algebraic geometry is represented by Peter Russell and Eyal Goren and is very much connected to the interests of other members of the department: Henri Darmon, Jacques Husrtubise and Niky Kamran.
Here we comment that an abelian variety is a complete algebraic group (over C it is topologically a torus) and the moduli space is a variety that parameterize all those abelian varieties.
www.math.mcgill.ca /goren/AGwebpage/webpage2.htm   (1584 words)

An algebraic map or regular map or morphism of quasiprojective varieties is a map of whose graph is closed.
An algebraic group is a group G in the category of quasiprojective varieties i.e.
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)

 Macaulay 2 home page
On the homotopy Lie algebra of an arrangement, by Graham Denham and Alexander I. Suciu, to appear in the Michigan Mathematical Journal, arXiv:math.AT/0502417.
Combinatorial secant varieties, by Bernd Sturmfels and Seth Sullivant, arXiv:math.AC/0506223.
On the ideals and singularities of secant varieties of Segre varieties, by J.M. Landsberg and Jerzy Weyman, arXiv:math.AG/0601452.
www.math.uiuc.edu /Macaulay2   (4711 words)

 The Math Forum - Math Library - Algebraic Geom.
An introduction to ordinary analytic geometry as studied in secondary school.
An area of algebraic geometry that deals with nonsingular curves of genus 1 - in English, solutions to equations y^2 = x^3 + A x + B. It has important connections to number theory and in particular to factorization of ordinary integers (and thus to cryptography).
An investigation of the Birch and Swinnerton-Dyer conjecture, which ties together the constellation of invariants attached to an abelian variety.
mathforum.org /library/topics/algebraic_g   (1930 words)

 Amazon.com: Algebraic Geometry: A First Course (Graduate Texts in Mathematics): Books: Joe Harris   (Site not responding. Last check: 2007-11-01)
The twisted cubic is given as the first example of a concrete variety that is not a hypersurface, along with their generalizations, the rational normal curves.
In addition, algebraic groups on varieties are discussed in lecture 10, allowing one to discuss a kind of glueing operation on varieties, just as in geometric topology, namely by taking the quotient of varieties via finite groups.
The behavior of a variety at a singular point is studied in lecture 20 using tangent cones.
www.amazon.com /exec/obidos/tg/detail/-/0387977163?v=glance   (2089 words)

 [No title]   (Site not responding. Last check: 2007-11-01)
If the group variety is affine, such a group is an extension of a connected linear group by a finite group.
This amounts to the statement that the connected component of the identity has finite index in G (since G is quasicompact), and that an affine group variety has a faithful matrix representation.
The quotient (for the existence there is something to prove!) should be an abelian variety and Spec(\Gamma) is an affine variety for which you can use the results posted before.
www.math.niu.edu /~rusin/known-math/99/alg_gp   (397 words)

 U of M Topics in Algebraic Geometry Seminar
An extremely explicit paper which shows how to apply the idea of the determinant of a complex to compute resultants.
In this paper, which is written in the lagnuage of combinatorics and polhedral geometry, the authors introduce a notion of fiber polytopes which generalizes secondary polytopes.
I will sketch the three main approaches the authors use to study and compute discriminants and resultants -- first through an application of the classical geometric notion of a dual variety, next through a tool from homological algebra known as the determinant of a complex, and finally via combinatorial methods involving triangulations of polytopes.
www.math.lsa.umich.edu /seminars/alggeo/topicsAG/topicsAGseminar.shtml   (727 words)

 GAP (singular) - Chapter 1: singular: the GAP interface to Singular
Singular is a Computer Algebra system for polynomial computations with emphasis on the special needs of commutative algebra, algebraic geometry, and singularity theory.
Singular is not executed in the current directory, but in a user-specified one, or in a temporary one.
Singular is ordered with respect to a term ordering (or, monomial ordering), that has to be specified together with the declaration of a ring.
www.gap-system.org /Manuals/pkg/singular/doc/chap1.html   (4537 words)

 Laura Taalman : Curriculum Vitae
An idea for a simple way to encourage students to read their textbooks and to take ownership of and responsibility for the material.
An investigation of 2-dimensional tesselations and 3-dimensional space-filling polyhedra and their use in modular architecture, in particular in the work of artist and architect Gregg Fleishman.
An article I wrote with Eugenie Hunsicker was one of two articles chosen by the MAA for this award, which honors authors of exceptional articles that are accessible to undergraduates and published in Math Horizons.
www.math.jmu.edu /~taal/CV2006.html   (1703 words)

 Amazon.com: Algebraic Geometry and Arithmetic Curves: Books: Qing Liu   (Site not responding. Last check: 2007-11-01)
This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves.
Singular curves are treated through a detailed study of the Picard group.
In conclusion, this book should be an invaluable resource to anyone who wishes to learn about schemes, especially with arithmetic applications in mind.
www.amazon.com /Algebraic-Geometry-Arithmetic-Curves-Qing/dp/0198502842   (1552 words)

By general position we mean that no three points lie on a line, no six points lie on a conic, and no eight lie points lie on a singular cubic with one of the eight points on the singularity.
The process of expressing the rational points on a variety as the union of images of rational points from other varieties.
The associated Prym variety is the connected component of the kernel of the Albanese map
www.aimath.org /WWN/qptsurface2/articles/html/4a   (1057 words)

 Abelian group actions on algebraic varieties with one fixed point   (Site not responding. Last check: 2007-11-01)
Theorem: Let X be a complete algebraic variety over an algebraically closed field of characteristic p>=0, and let G be a finite abelian group acting on X. Assume the order of G is l^r, where l is a prime different from p.
If the fixed point set consists of exactly one point x, then X is singular in X. Corollary: If X is smooth then G has either no fixed point or at least two of them.
This is an algebraic analogue of (much deeper) results of Conner-Floyd, Atiyah-Bott and others in the topological category.
www.math.rutgers.edu /~knop/papers/Amir.html   (123 words)

 Singular Manual: Elimination
,but even if this map germ is finite, we are in general not able to compute the image germ because for this we would need an implementation of the Weierstrass preparation theorem.
This is achieved by an ordering with the block of t-variables having a global ordering (and preceding the x-variables) and the x-variables having a local ordering.
In any case, if the input is weighted homogeneous (=quasihomogeneous), the weights given to the variables should be chosen accordingly.
www.math.lsu.edu /singular/sing_475.htm   (762 words)

 NSDL Metadata Record -- Algebraic Curve -- from MathWorld
A K-rational point is a point (X, Y) on the curve, where X and Y are in the field K. See also: Algebraic Geometry, Algebraic Variety, Curve
Arbarello, E.; Cornalba, M.; Griffiths, P. A.; and Harris, J. Geometry of Algebraic Curves, I. New York: Springer-Verlag, 1985.
Coolidge, J. A Treatise on Algebraic Plane Curves.
nsdl.org /mr/697642   (82 words)

 The Denef-Loeser series for toric surface singularities, Monique Lejeune-Jalabert, Ana J. Reguera
Let $H$ denote the set of formal arcs going through a singular point of an algebraic variety $V$ defined over an algebraically closed field $k$ of characteristic zero.
Recently (1999), J. Denef and F. Loeser have proved that the Poincar\'{e} series associated with the image of $j^s(H)$ in some suitable localization of the Grothendieck ring of algebraic varieties over $k$ is a rational function.
We compute this function for normal toric surface singularities.
projecteuclid.org /getRecord?id=euclid.rmi/1063050167   (212 words)

 International Conference on Symbolic and Algebraic Computation
A new algorithm for the geometric decomposition of a variety
A Hadamardtype bound on the coefficients of a determinant of polynomials.
An algorithm to conlput(; the exponential part of a forinal fundamc.ntal matrix solution of a lin('ar diffcr(-'ntial system.
physjob.nudl.org /~akrowne/acm_sample_in_citidel_format.xml   (4456 words)

 Citebase - Residue forms on singular hypersurfaces
The purpose of this paper is to point out a relation between the canonical sheaf and the intersection complex of a singular algebraic variety.
Let M be a complex manifold, X⊂ M a singular hypersurface.
Applying resolution of singularities sometimes we are able to construct residue classes either in L
citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0301313   (174 words)

 Singular: Overview
SINGULAR's development started in 1986 and it has by now become one of the most powerful and efficient systems for polynomial computations.
Here are some of the most important features of SINGULAR:
Extensive libraries of procedures, written in the SINGULAR language
www.singular.uni-kl.de /overview.html   (133 words)

 Seminar KULeuven-UGent, Number Theory and Algebraic Geometry
K.U.Leuven, Department of Mathematics, research unit algebraic geometry and number theory.
An assymptotic vanishing theorem for unions of multiple points on a projective algebraic variety
A geometric approach to the study of Severi varieties on surfaces
cage.rug.ac.be /~kulrug/webpagina.html   (1676 words)

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