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Topic: Algebraic surfaces

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	Algebraic surface - Wikipedia, the free encyclopedia
		In the case of geometry over the complex number field, an algebraic surface is therefore of complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.
		The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two).
		The birational geometry of algebraic surfaces is rich, because of blowing up (also known as a monoidal transformation); under which a point is replaced by the curve of all limiting tangent directions coming into it (a projective line).
	en.wikipedia.org /wiki/Algebraic_surface (396 words)

	Surface - Wikipedia, the free encyclopedia
		In mathematics (topology), a surface is a two-dimensional manifold.
		The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation.
		This notion of a surface is distinct from the notion of an algebraic surface.
	en.wikipedia.org /wiki/Surface (639 words)

	Algebraic geometry - Wikipedia, the free encyclopedia
		Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry.
		Algebraic geometry was developed largely by the Italian geometers in the early part of the 20th century.
		Commutative algebra (earlier known as elimination theory and then ideal theory, and refounded as the study of commutative rings and their modules) had been and was being developed by David Hilbert, Max Noether, Emanuel Lasker, Emmy Noether, Wolfgang Krull, and others.
	en.wikipedia.org /wiki/Algebraic_geometry (1802 words)

	Algebraic geometry -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of (A mathematical expression that is the sum of a number of terms) polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations.
		In classical algebraic geometry, this field was always C, the complex numbers, but many of the same results are true if we assume only that k is (Click link for more info and facts about algebraically closed) algebraically closed.
		Algebraic geometry was developed largely by the (Click link for more info and facts about Italian geometers) Italian geometers in the early part of the 20th century.
	www.absoluteastronomy.com /encyclopedia/a/al/algebraic_geometry.htm (2010 words)

	K3 surface
		A non-singular quartic surface in projective space of three dimensions is a K3 surface.
		The definition used in algebraic geometry is that the canonical class is trivial, and H
		A '\Kummer surface' is the quotient of a two-dimensional abelian variety A by the action of a → −a.
	www.brainyencyclopedia.com /encyclopedia/k/k3/k3_surface.html (276 words)

	Basic Concepts
		The highest degree of all terms is the degree of the algebraic surface.
		Therefore, spheres and all quadric surfaces are algebraic surfaces of degree two, while torus is a degree four algebraic surface.
		The left surface is a hyperbolic paraboloid and the right one is a ring Dupin cyclide which is a degree 4 rational surface.
	www.cs.mtu.edu /~shene/COURSES/cs3621/NOTES/surface/basic.html (1555 words)

	Cubic surfaces
		An algebraic surface is one of the form f(x,y,z) = 0 where f(x,y,z) is a polynomial in x, y and z.
		The surface whose history we are interested in for this short article is a surface of order three which is called a cubic surface.
		He divided cubic surfaces into 23 species according to the nature of their singularities in 1863 and he published the classification in his paper On the distribution of surfaces of the third order into species, in reference to the presence or absence of singular points and the reality of their lines.
	www-groups.dcs.st-and.ac.uk /~history/PrintHT/Cubic_surfaces.html (1002 words)

	Examples of Algebraic Surfaces
		The reason why the above singularities are important is that when you have a family of surfaces controlled by a number of parameters you will often find some surfaces which contain one of these singularities.
		Perhaps the most fun is had when the value of c is changed and a surfaces with three A1 points is displayed.
		Dupin Cyclides This is a family of quartic surfaces which are generalization of a torus.
	www.javaview.de /services/algebraic/examples.html (956 words)

	ipedia.com: Algebraic geometry Article (Site not responding. Last check: 2007-11-06)
		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.
		While projective geometry was originally established on a synthetic foundation, the use of homogenous co-ordinates allowed the introduction of algebraic techniques.
		Commutative algebra (as the study of commutative rings and their ideals) had been and was being developed by David Hilbert, Max Noether, Emanuel Lasker, Emmy Noether, Wolfgang Krull, and others.
	www.ipedia.com /algebraic_geometry.html (1723 words)

	Help for the Sing Surf program
		This will load a new surface into the viewing window and the control panel will be loaded with panels for controlling this type of curve and surface.
		Examples of these can be found in the help pages for the individual programs algebraic surfaces algebraic curves algebraic curves in 3D parameterised surfaces parameterised curves implicit surfaces.
		The algorithm for generating algebraic surfaces is a little complicated and is fully described in my paper A new method for drawing Algebraic Surfaces.
	www.singsurf.org /singsurf/help.html (2097 words)

	14J: Surfaces and higher-dimensional varieties
		Surfaces are often studied by considering maps from or to algebraic curves -- that is, by studying embedded (or immersed) curves, or by viewing the surface as fibred over a curve.
		A surface may be presented as fibred over a curve, with fibres of different genera.
		Parameterizing the family of lines tangent to two spheres (an algebraic surface).
	www.math.niu.edu /~rusin/known-math/index/14JXX.html (510 words)

	Citations: Ray tracing algebraic surfaces - Hanrahan (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		The surface normal is then computed using the display model (voxels, polygons, or by sampling the field value in small increments in each direction.
		....surfaces to another representation which is easier to display such as polygon meshes [1] 5] 10] 15] or parametric representations [16] The second approach to rendering of implicit surfaces is to ray trace them.
		Algebraic implicit surfaces produce a polynomial g(t) for which root finding is easier.
	citeseer.ist.psu.edu /context/95825/0 (2274 words)

	Computation of Adjoints for Surfaces
		The `m-adjoints of degree n' of the surface.
		An `m-adjoint of degree n' of a surface is a polynomial of degree n that vanishes with a certain order (depending on m) on the singularities of the surface.
		e are given an algebraic surface by its equation, and we want to find its parametrization in terms of rational functions in two parameters, if such a representation exists.
	www.risc.uni-linz.ac.at /projects/basic/adjoints (700 words)

	Algebraic surfaces
		A Tritangent of a cubic surface is a plane which intersects the cubic surface in the union of three lines (instead of a irreducible cubic curve), and the three intersection points of the three lines are then tangent points, hence the name.
		There are several special kinds of quartic surfaces, the Kummer surfaces, named after W. Kummer, the symmetroids, which are the zero locus of the determinant of a symmetric 4 x 4 matrix of linear forms, and the desmic surfaces, so-called because of their relation to desmic tetrahedra.
		As far as the desmic surfaces are concenrned, these are more special than the symmetroids: there is just a one-dimensional family of them, and they have 12 ordinary double points, which are the vertices of three tetrahedra in three-space.
	www.mathematik.uni-kl.de /~hunt/drawings.html (1017 words)

	Amazon.com: Books: Principles of Algebraic Geometry (Site not responding. Last check: 2007-11-06)
		This book would probably be one the most useful one for those interested in applications, for it is an overview of algebraic geometry from the complex analytic point of view, and complex analysis is a subject that most engineers and scientists have had to learn at some point in their careers.
		Algebraic geometry is an exciting subject, but one must master some background material before beginning a study of it.
		The study of surfaces is studied in chapter 4, with the differences between its study and the theory of curves (Riemann surfaces) emphasized.
	www.amazon.com /exec/obidos/tg/detail/-/0471050598?v=glance (2462 words)

	Amazon.com: Books: Algebraic Geometry (Graduate Texts in Mathematics) (Site not responding. Last check: 2007-11-06)
		That algebraic geometry has so many applications is quite amazing, since it was not too long ago that it was thought of as a highly abstract, esoteric topic.
		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 (2618 words)

	Animated algebraic surfaces (Site not responding. Last check: 2007-11-06)
		Deformation of hyperboloid, which deforms from one sheet into two sheets, with a cone at the middle of the deformation (when the two sheets touch each other).
		We arrive at singular surfaces at an intermediate stage as well as at the end.
		Riemann surface in 2-1 cover over the plane, with two ramification points.
	wims.unice.fr /~xiao/gallery (312 words)

	Citebase - Algebraic surfaces holomorphically dominable by C^2 (Site not responding. Last check: 2007-11-06)
		In this paper, we attempt to classify algebraic surfaces X which are dominable by C 2 using a combination of techniques from algebraic topology, complex geometry and analysis.
		Using the Kodaira dimension and the fundamental group of X, we succeed in classifying algebraic surfaces which are dominable by C 2 except for certain cases in which X is an algebraic surface of Kodaira dimension zero and the case when X is rational without any logarithmic 1-form.
		With the exceptions noted above, we show that for any algebraic surface of Kodaira dimension less than 2, dominability by C 2 is equivalent to the apparently weaker requirement of the existence of a holomorphic image of C which is Zariski dense in the surface.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:math/9903193 (7547 words)

	OUP: Open Algebraic Surfaces: Miyanishi (Site not responding. Last check: 2007-11-06)
		Open algebraic surfaces are a synonym for algebraic surfaces that are not necessarily complete.
		An open algebraic surface is understood as a Zariski open set of a projective algebraic surface.
		The theory of open algebraic surfaces is applicable not only to algebraic geometry, but also to other fields, such as commutative algebra, invariant theory, and singularities.
	www.oup.co.uk /isbn/0-8218-0504-5 (343 words)

	Amazon.ca: Books: Algebraic Surfaces (Site not responding. Last check: 2007-11-06)
		The main aim of this book is to present a completely algebraic approach to the Enriques' classification of smooth projective surfaces defined over an algebraically closed field of arbitrary characteristic.
		This algebraic approach is one of the novelties of this book among the other modern textbooks devoted to this subject.
		The book can be useful as a textbook for a graduate course on surfaces, for researchers or graduate students in algebraic geometry, as well as those mathematicians working in algebraic geometry or related fields.
	www.amazon.ca /exec/obidos/ASIN/0387986685 (257 words)

	Amazon.ca: Books: Algebraic Surfaces and Holomorphic Vector Bundles (Site not responding. Last check: 2007-11-06)
		It is aimed at graduate students who have had a thorough first year course in algebraic geometry (at the level of Hartshorne's ALGEBRAIC GEOMETRY), as well as more advanced graduate students and researchers in the areas of algebraic geometry, gauge thoery, or 4-manifold topolgogy.
		A novel feature of the book is its integrated approach to algebraic surface theory and the study of vector bundle theory on both curves and surfaces.
		Thus vector bundles over curves are studied to understand ruled surfaces, and then reappear in the proof of Bogomolov's inequality for stable bundles, which is itself applied to study canonical embeddings of surfaces via Reider's method.
	www.amazon.ca /exec/obidos/ASIN/0387983619 (400 words)

	3-D Surface Segmentation of Free-Form Objects using Implicit Algebraic Surfaces (Site not responding. Last check: 2007-11-06)
		A free-form surface is one with a well defined surface normal that is continuous everywhere except at vertices, edges and cusps.
		The authors wish to derive a surface description of objects that may vary in shape and complexity without any restriction on the typeof surfaces on the object.
		This paper presents a surface representation scheme that uses edge information to build a surface description using algebraic implicit surfaces.
	www.stanford.edu /~ctj/liteseer/segparmf/benlamri03surface (323 words)

	Riemann surfaces, plane algebraic curves and their period matrices (Site not responding. Last check: 2007-11-06)
		The aim of this paper is to present theoretical basis for computing a representation of a compact Riemann surface as an algebraic plane curve and to compute a numerical approximation for its period matrix.
		The contribution of the present paper is the design of an algorithm which is based on the classical results and computes first an approximation of a polynomial representing a given compact Riemann surface as a plane algebraic curve and further computes an approximation for a period matrix of this curve.
		This problem was first solved, in the case of symmetric Riemann surfaces, in Seppälä, Mika: Computation of period matrices of real algebraic curves.-Discrete Comput Geom 11:65-81 (1994).
	www.math.fsu.edu /~seppala/papers/PeriodMatrices/PeriodMatrices.html (252 words)

	Coombes: Other Projects
		The following notes are from the beginning of an algebraic geometry course I taught.
		The following notes are reports on talks given in an informal seminar on the classification of algebraic surfaces over an arbitrary field.
		Having struggled mightily on several occassions to convey the ideas of algebraic geometry to a class of graduate students, I've become convinced that there is no good linear way to present this material.
	odin.mdacc.tmc.edu /~krc/math/proj.html (497 words)

	Algebraic Number Theory Archive (Site not responding. Last check: 2007-11-06)
		math.NT/0409352: 20 Sep 2004, Abelian surfaces of GL2-type as Jacobians of curves, by Josep Gonzalez, Jordi Guardia, Victor Rotger.
		ANT-0212: 15 Nov 1999, On the Andre-Oort conjecture for Hilbert modular surfaces, by Bas Edixhoven.
		ANT-0188: 15 Jun 1999, The Mordell-Weil rank of the Jacobian of a curve of genus 2 with multiplication by a square root of 2, by Peter R. Bending.
	front.math.ucdavis.edu /ANT (12251 words)

	Uni Gàttingen, Mathematisches Institut: Homepage Mustermann
		"Moduli of algebraic surfaces", in "Theory of moduli", Proc.
		"Algebraic Geometry" 1990, Springer LNM 1502, 51-70 (1992).
		"Singular bidouble covers and the construction of interesting algebraic surfaces", 31 pages, to appear in the Proceedings of the Conference in honour of F. Hirzebruch's 70th Birthday, A.M.S. Contemp.
	www.uni-math.gwdg.de /catanese (1259 words)

	Algebraic Surfaces on the Web
		A VRML model of the surface you entered should be returned.
		This program has been adapted from the Algebraic Surfaces module of the Liverpool Surface Modelling Package which allows mayny different types of mathematical curves and surfaces to be visulised.
		If a degenerate surface is specified the program can take a very long time to run.
	www.amsta.leeds.ac.uk /~rjm/lsmp/asurfcgi.html (423 words)

	DC MetaData for:Holomorphic vector bundles on non-algebraic surfaces
		The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open.
		We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII.
		Our methods, which are based on Donaldson theory and deformation theory, can be used to solve the existence problem of holomorphic vector bundles on further classes of non-algebraic surfaces.
	www.mathematik.uni-osnabrueck.de /preprints/shadow/calg0111.rdf.html (86 words)

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