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Topic: Birational geometry

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	Birational geometry (Site not responding. Last check: 2007-10-07)
		In mathematics, birational geometry is a part of thesubject of algebraic geometry, that deals with the geometry ofan algebraic variety that is dependent only on its function field.
		In the case of dimension 2, the birational geometry ofalgebraic surfaces was largely worked out by the Italian school of algebraic geometry, two decades on either side of the year 1900.From about 1970 advances have been made, giving a good theory of birational geometry for dimension 3.
		One of the first results in the subject is the birational isomorphism of the projective plane, and a non-singular quadric Q in projective3-space.
	www.therfcc.org /birational-geometry-69628.html (316 words)

	Birational geometry - InformationBlast
		] In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field.
		In the case of dimension 2, the birational geometry of algebraic surfaces was largely worked out by the Italian school of algebraic geometry, two decades on either side of the year 1900.
		One of the first results in the subject is the birational isomorphism of the projective plane, and a non-singular quadric Q in projective 3-space.
	www.informationblast.com /Birational_geometry.html (330 words)

	Category:Geometry - Wikipedia, the free encyclopedia
		Geometry is the branch of mathematics dealing with spatial relationships.
		From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry.
		Such axioms are insusceptible to proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions.
	en.wikipedia.org /wiki/Category:Geometry (115 words)

	Federigo Enriques - Wikipedia, the free encyclopedia
		The Enriques classification, of complex algebraic surfaces up to birational equivalence, was into five main classes, and was background to further work until Kodaira reconsidered the matter in the 1950s.
		There remains the class of elliptic surfaces, which are fiber bundles over a curve with elliptic curves as fiber, having a finite number of modifications (so there is a bundle that is locally trivial actually over a curve less some points).
		The question of classification is to show that any surface, lying in projective space of any dimension, is in the birational sense (after blowing up and blowing down of some curves, that is) accounted for by the models already mentioned.
	en.wikipedia.org /wiki/Federigo_Enriques (457 words)

	Birational geometry (Site not responding. Last check: 2007-10-07)
		Such transformations, given by rational functions in the co-ordinates, can be undefined not just at isolated points on curves, but on entire curves on a surface, and so on.
		This has to be understood in the extended sense that the composition, in either order, is only in fact defined on a non-empty Zariski open subset.
		An example is the Cremona group of birational automorphisms of the projective plane.
	www.tocatch.info /en/Birational_geometry.htm (380 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		Inequalities on log-canonical threshold and birational geometry of Fano hypersurfaces,
		Arithmetic and Algebraic Geometry Conference, University of Tokyo, January 2004.
		Conference in Algebraic Geometry in memory of Paolo Francia, Università di Genova, September 2001.
	www.math.lsa.umich.edu /~defernex/lectures.html (475 words)

	14: Algebraic geometry
		Algebraic geometry combines the algebraic with the geometric for the benefit of both.
		Conversely, the geometry of sets defined by equations is studied using quite sophisticated algebraic machinery.
		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)

	Algebraic surface (Site not responding. Last check: 2007-10-07)
		In the case of geometry over the complex number field, an algebraic surface is therefore of complex dimension two (as a complex manifold) and so of dimension four as a smooth manifold.
		That is, for example, a cubic surface has a function field isomorphic to that of the projective plane, being the rationalfunctions in two indeterminates.
		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 alllimiting tangent directions coming into it (a projective line).Certain curves may also be blown down, but there is a restriction (self-intersection number must be −1).
	www.therfcc.org /algebraic-surface-210502.html (265 words)

	Higher Dimensional Complex Geometry
		This idea goes back to the treatment of conic sections by the Greeks and non-Euclidean geometries in the 19th century, but it is only in the past 20 years that a general picture has emerged in the context of higher-dimensional algebraic geometry.
		Geometry, and especially algebraic geometry, is increasingly the language of theoretical physics and string theory.
		In addition, Kachi and Takagi, young researchers in the field, gave introductory lectures on birational geometry in general, and on the Shokurov approach to flips in particular.
	www.newton.cam.ac.uk /reports/0102/hdg.html (2761 words)

	Birational Geometry of Algebraic Varieties (Cambridge Tracts in Mathematics) by Janos Kollár [ISBN: 0521632773] - Find ...
		Birational Geometry of Algebraic Varieties (Cambridge Tracts in Mathematics)
		Some level of familiarity with standard algebraic geometry (like what're written in Hartshorne's famous book) is required to be able to read this book.
		It may not be very reader-friendly, but it's a book that you must read if you want to study birational geometry.
	www.gettextbooks.com /isbn_0521632773.html (206 words)

	ARCC Workshop: Compact moduli spaces and birational geometry (Site not responding. Last check: 2007-10-07)
		Perhaps the first example is the Deligne/Mumford compactification of the moduli space of stable curves, where the limiting curves are dictated by the structure of canonical models for surfaces fibered over curves.
		This was extended to surfaces by Koll'ar/Shepherd-Barron and Alexeev, which led to work of Corti, Hacking, Tevelev/Keel, Alexeev, and others, where birational geometry inspired the choice of limiting objects, and sometimes played a role in constructing moduli spaces.
		The main goals of this workshop are: to promote cross-fertilization by bringing together specialists in birational geometry and moduli theory; to make the techniques of the field more widely-known and accessible; and to identify concrete, tractable questions for young researchers entering the area.
	aimath.org /ARCC/workshops/birational.html (342 words)

	Linear Algebraic Groups and Related Structures (Site not responding. Last check: 2007-10-07)
		The aim of this paper is to investigate the birational geometry of Generalized Severi-Brauer varieties.
		A conjecture of Amitsur states that two Severi-Brauer varieties $V(A)$ and $V(B)$ are birational if the underlying algebras $A$ and $B$ are the same degree and generate the same cyclic subgroup of the Brauer group.
		We present a generalization of this conjecture to Generalized Severi-Brauer varieties, and show that in many cases we may reduce the new conjecture to the case where every subfield of the algebras is maximal, and in particular to the case where the algebras have prime power degree.
	www.math.uni-bielefeld.de /LAG/man/082.html (165 words)

	Algebraic Geometry and Commutative Algebra in Helsinki
		By now there are several related techniques which go by the name of motivic integration and many applications of these to problems in fields as various as number theory, algebraic geometry and representation theory.
		The course begins with an introductory lecture, which is meant for a general mathematician.
		The three subsequent lectures require good mastering of the techniques of algebraic geometry.
	www.helsinki.fi /~ehyry/motivic.html (184 words)

	2005-1 (Site not responding. Last check: 2007-10-07)
		The Schhol is mainly aimed to Phd students and young researchers in Algebraic Geometry, introducing the participants to research, beginning from a basic level with a view towards the applications and to the most recent results.
		The Cremona group of the birational transformations of a projective n-dimensional space is, still now, not very much understood, at least if n is not lesser than 3.
		The relations with the modern birational geometry will be discussed as well various related topics, among them: homaloidal hypersurfaces, special linear systems of plane curves, congruences of rational curves and their birational classification, rationality problems for conic bundles, etcetera.
	calvino.polito.it /~geometri/2005-1.htm (173 words)

	David R. Morrison: Publications
		(with R. Friedman), The birational geometry of degenerations: An overview, The Birational Geometry of Degenerations (R. Friedman and D. Morrison, eds.), Progress in Math., vol.
		On the moduli of Todorov surfaces, Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata (H. Hijikata et al., eds.), vol.
		The geometry underlying mirror symmetry, New Trends in Algebraic Geometry (K. Hulek, F. Catanese, C. Peters, and M. Reid, eds.), London Math.
	www.cgtp.duke.edu /~drm/publications.html (1260 words)

	Classification and birational geometry
		The Sarkisov program, factoring birational maps between Mfs into elementary steps, has led to a recent breakthrough in distinguishing rigid from nonrigid 3-folds, thus contributing to classical questions such as criteria for irrationality and the Cremona group.
		It is striking that his proof uses the Thaddeus-Mumford principle of geometric invariant theory, a method devised for applications to moduli theory.
		odarczyk, Birational cobordisms and factorization of birational maps, math.AG/9904074, 23 pp.
	euclid.mathematik.uni-kl.de /NEW/node27.html (378 words)

	Definition of Algebraic function field
		In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions.
		In the particular case of an algebraic curve C, that is, dimension 1, it follows that any two non-constant functions F and G on C satisfy a polynomial equation P(F,G) = 0.
		Properties of the variety V that depend only on the function field are studied in birational geometry.
	www.wordiq.com /definition/Algebraic_function_field (218 words)

	Birational Geometry of Algebraic Varieties - Cambridge University Press
		Birational Geometry of Algebraic Varieties - Cambridge University Press
		$75.00 (C) One of the major discoveries of the past two decades in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties.
		It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.
	www.cup.cam.ac.uk /us/catalogue/catalogue.asp?isbn=0521632773&ss=sam (215 words)

	MORI, SHIGEFUMI - CIRS (Site not responding. Last check: 2007-10-07)
		Professor of Algebraic Geometry at the Research Institute of Mathematical Sciences (RIMS), Kyoto University, Kyoto, Japan.
		Algebraic Geometry, especially birational classification and the birational geometry of algebraic varieties.
		Mori has worked on algebraic manifolds with ample tangent bundles and was the first to prove the Hartshorne conjecture in 1978.
	www.cirs-tm.org /researchers/researchers.php?id=307 (152 words)

	UM Mathematics: Faculty-Detail
		My interests primarly focus on birational geometry of higher dimensional varieties.
		I am also interested in multiplier ideals and in geometric properties that can be deduced by the study of rational curves on varieties.
		I have done some research on automorphisms and birational transformations of surfaces and on certain classification problems concerning regular sections of ample vector bundles.
	www.math.lsa.umich.edu /people/facultyDetail.php?id=333 (87 words)

	Categorical Geometry Homepage
		[4] Factorization of birational morphisms of regular schemes.
		Algebraic geometry and algebraic number theory (Tianjin, 1989--1990), 77--91, Nankai Ser.
		[6] (with Tie Luo) Factorization of birational morphisms with fiber dimension bounded by 1.
	www.geometry.net /cg/Luohomepage.html (68 words)

	Visitors (Site not responding. Last check: 2007-10-07)
		Of particular interest is the isomorphic classification as well as in the geometry of Banach spaces.
		The geometry of 4-manifolds, the Seiberg-Witten invariants, and Conjugate Connections.
		Algebraic Geometry, in particular curves on threefolds and birational geometry.
	www.math.okstate.edu /undergrad/handbook/node77.html (160 words)

	Explicit Birational Geometry of 3-folds (London Mathematical Society Lecture Note Series) New, Used Books, Cheap ...
		This volume is an integrated suite of papers centred around applications of Mori theory to birational geometry.
		Explicit Birational Geometry of 3-Folds (By Alessio Corti (Editor))
		Birational Geometry of Algebraic Varieties (By Janos Kollar)
	www.bookfinder4u.com /detail/0521636418.html (179 words)

	INI Programme HDG (Site not responding. Last check: 2007-10-07)
		Classification: problems on existence and moduli of algebraic surfaces and 3-folds, including methods of projective and birational geometry, commutative algebra, toric geometry, etc. The minimal model program, flips and birational contractions.
		Resolution of quotient singularities, McKay correspondence and stringy geometry.
		Algebra, number theory, physics, gauge theory, symplectic geometry and other aspects of differential geometry, hyperKhler and related "special" geometries.
	www.newton.cam.ac.uk /programs/HDG (145 words)

	WAGS-Spring 2003: Schedule (Site not responding. Last check: 2007-10-07)
		The computaion is done by means of the so-called tropical algebraic geometry.
		The answer is presented as a number of certain lattice paths in the corresponding convex polygon.
		The dimension of the curves in the count is equal to the length of the corresponding lattice paths.
	math.stanford.edu /%7Eazinger/wags03sp/schedule.html (398 words)

	Red Harvest by Dashiell Hammett, ISBN 0679722610 And Explicit Birational Geometry of 3-Folds by Alessio Corti, ISBN ... (Site not responding. Last check: 2007-10-07)
		One of the main achievements of algebraic geometry over the past twenty years is the work of Mori and others extending minimal models and the Enriques-Kodaira classification to 3-folds.
		This integrated suite of papers centers around applications of Mori theory to birational geometry.
		These contributions work for the first time with a representative class of Fano varieties, 3-fold hypersurfaces in weighted projective space, and they include an attractive introductory treatment with a wealth of detailed computation of special cases.
	wstevenash.com /harvest.htm (224 words)

	Thanksgiving Day And Birational Geometry of Algebraic Varieties by Janos Kollar, ISBN 0521632773
		Introduces Thanksgiving Day, explaining why we eat turkey, when the first Thanksgiving took place, and why Abraham Lincoln thought Thanksgiving would be good for the United States.
		Birational Geometry of Algebraic Varieties by Janos Kollar, ISBN 0521632773
		One of the major discoveries of the past two decades in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties.
	bricktannia.com /thanksgiving.htm (184 words)

	3-fold links (Site not responding. Last check: 2007-10-07)
		Corti and M. Mella, Birational geometry of terminal quartic 3-folds,
		Reid, Graded rings and birational geometry, in Proc.
		Geometry, In memory of Paolo Francia (Genova, Sep 2001),
	www.maths.warwick.ac.uk /~miles/3folds (231 words)

	Library
		Griffiths, P. & Harris, J Principles of algebraic geometry.
		Thaddeus, M. An introduction to the topology of the moduli space of stable bundles on a Riemann surface.
		Geometry and physics (Aarhus, 1995), 71-99, Lecture Notes in Pure and Appl.
	www.bath.ac.uk /~masgks/ShortCourse/library.html (350 words)

	Ivan Cheltsov
		SEMINAR ON ALGEBRAIC GEOMETRY, HUMBOLT UNIVERSITY AT BERLIN, 2004
		SEMINAR ON ALGEBRAIC GEOMETRY, HUMBOLT UNIVERSITY AT BERLIN, 1999
		SEMINAR ON ALGEBRAIC GEOMETRY, HUMBOLT UNIVERSITY AT BERLIN, 1997
	www.math.uga.edu /~cheltsov/gab.html (301 words)

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