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1. Algebraic Geometry - Wikipedia, The Free Encyclopedia
algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra,
http://en.wikipedia.org/wiki/Algebraic_geometry
##### Algebraic geometry
Jump to: navigation search Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra , especially commutative algebra , with the language and the problematics of geometry . It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis topology and number theory . Initially a study of polynomial equations in many variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations, as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique. The fundamental objects of study in algebraic geometry are algebraic varieties , geometric manifestations of solutions of systems of polynomial equations Plane algebraic curves , which include lines circles parabolas lemniscates , and Cassini ovals , form one of the best studied classes of algebraic varieties. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve relative position of different curves and relations between the curves given by different equations.

2. Algebraic Geometry -- From Wolfram MathWorld
algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of
http://mathworld.wolfram.com/AlgebraicGeometry.html
 Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Less... Algebraic Geometry Algebraic geometry is the study of geometries that come from algebra, in particular, from rings . In classical algebraic geometry , the algebra is the ring of polynomials , and the geometry is the set of zeros of polynomials, called an algebraic variety . For instance, the unit circle is the set of zeros of and is an algebraic variety , as are all of the conic sections In the twentieth century, it was discovered that the basic ideas of classical algebraic geometry can be applied to any commutative ring with a unit, such as the integers . The geometry of such a ring is determined by its algebraic structure, in particular its prime ideals . Grothendieck defined schemes as the basic geometric objects, which have the same relationship to the geometry of a ring as a manifold to a coordinate chart . The language of category theory evolved at around the same time, largely in response to the needs of the increasing abstraction in algebraic geometry. As a consequence, algebraic geometry became very useful in other areas of mathematics, most notably in

3. Algebraic Geometry
algebraic geometry. pdf (for printing on letter paper 11x8.5 in) pdf (for printing on a4 paper 29.7x21 cm) The page numbering etc. for the two files is
http://www.jmilne.org/math/CourseNotes/math631.html
##### Algebraic Geometry
pdf (for printing on letter paper 11x8.5 in)
pdf (for printing on a4 paper 29.7x21 cm)

The page numbering etc. for the two files is identical. v2.01 (August 24, 1996). First version on the web.
v3.01 (June 13, 1998). Added 5 sections (25 pages) and an index. Minor changes to Sections 0-8. 157pp.
v4.00 (October 30, 2003). Fixed errors; many minor revisions; added exercises; added two sections; 206 pages.
v5.00 (February 20, 2005). Heavily revised; most numbering changed; 227 pages.
##### Contents
• Preliminaries
• Algebraic sets Affine algebraic varieties Algebaic varieties Local study Projective varieties Complete varieties Finite maps Dimension theory Regular maps and their fibres Algebraic spaces: geometry over an arbitrary field Divisors and intersection theory Coherent sheaves; invertible sheaves Differentials (Outline) Algebraic varieties over the complex numbers (Outline) Descent Theory Lefschetz Pencils (Outline)
• Solutions to the Exercises
Annotated Bibliography
Index Errata

4. Algebraic Geometry
Some information about algebraic geometry and geometers.
http://www.math.byu.edu/~jarvis/alg-geom.html
Search BYU
##### Algebraic Geometry
Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate. David Mumford
##### Lecture notes and videos
• MSRI lectures High quality streaming video of many lectures on a wide range of mathematical topics. PDF slides and notes also available.

5. Front: Math.AG Algebraic Geometry
arXiv0801.2834 Relative Proportionality for subvarieties of moduli spaces of K3 and abelian surfaces. S. MüllerStach, E. Viehweg, K. Zuo. math.AG.
http://front.math.ucdavis.edu/math.AG
 Front for the arXiv Fri, 14 Mar 2008 Front math AG search register submit journals ... iFAQ math.AG Algebraic Geometry Calendar Search Atom feed Search Author Title/ID Abstract+ Category articles per page Show Search help Recent New articles (last 12) 14 Mar arXiv:0803.2026 Geometry of obstructed equisingular families of projective hypersurfaces. Anna Gourevitch , Dmitry Gourevitch math.AG 14 Mar arXiv:0803.1996 Rational points on homogeneous varieties and Equidistribution of Adelic periods. Alex Gorodnik , Hee Oh (with appendix by Mikhail Borovoi). math.AG math.NT 13 Mar arXiv:0803.1825 On the algebraic geometry of polynomial dynamical systems. Abdul S. Jarrah , Reinhard Laubenbacher math.AG math.AC 13 Mar arXiv:0803.1756 Equisingularity of families of hypersurfaces and applications to mappings. Kevin Houston math.AG math.CV 13 Mar arXiv:0803.1743 Poincare series of filtrations corresponding to ideals on surfaces. A. Campillo (Valladolid University, Spain), F. Delgado (Valladolid University, Spain), S. M. Gusein-Zade (Moscow State University, Russia). math.AG

6. Algebraic Geometry Authors/titles Recent Submissions
Subjects algebraic geometry (math.AG); Representation Theory (math.RT) AG/0509587 with a not poor exposition and no essential material added.
http://arxiv.org/list/math.AG/recent
##### arXiv.orgmathmath.AG
Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages
##### Authors and titles for recent submissions
[ total of 28 entries:
[ showing 25 entries per page: fewer more all
##### Fri, 14 Mar 2008
arXiv:0803.2026 ps pdf other
Title: Geometry of obstructed equisingular families of projective hypersurfaces Authors: Anna Gourevitch Dmitry Gourevitch Comments: 26 pages Subjects: Algebraic Geometry (math.AG)
arXiv:0803.1996 ps pdf other
Title: Rational points on homogeneous varieties and Equidistribution of Adelic periods Authors: Alex Gorodnik Hee Oh (with appendix by Mikhail Borovoi) Comments: 58 pages, and 27 pages of appendix Subjects: Algebraic Geometry (math.AG) ; Number Theory (math.NT)
arXiv:0803.1950 (cross-list from math.CV) [ ps pdf other
Title: Capacities and weighted volumes of line bundles Authors: Robert Berman Sebastien Boucksom Comments: 57 pages, 1 figure Subjects: Complex Variables (math.CV) ; Algebraic Geometry (math.AG); Number Theory (math.NT)

7. Algebraic Geometry
For doing calculations in algebraic geometry and commutative algebra, Grayson and Stillman s Macaulay2 program is more powerful than Maple or Mathematica.
http://www.math.purdue.edu/~dvb/algeom.html
##### Introduction to Algebraic Geometry
Math 665, spring 06 Donu Arapura Blow up
• Examples
• Geometry of 2x2 nilpotent matrices
• Blowing up a cusp ...
• Books
##### Examples
In order to get a feeling for what algebraic geometry is, let's to go through some simple examples.
##### Geometry of 2x2 nilpotent matrices
Consider the space of 2x2 matrices over a field k with trace 0. These can be parameterized by 3-tuples (x,y,z) where x, y and z appear in the 11, 12, and 21 positions; therefore this can be identified with three dimensional affine space A . In these coordinates the determinant det(x,y,z) = -x -yz. The determinant is an regular map A A . The fibers X t =det are algebraic subsets of A . These are in fact varieties since the polynomials -x -yz-t are irreducible for each t in k. Let's study the geometry of these sets. Suppose k= C , then I claim that X t is isomorphic to X whenever t is nonzero. To see this, choose a such that a isomorphism between X and X t ; this can be checked by comparing coordinate rings. ( This works even if C is replaced by an algebraically closed field. But it fails in general. For example when k =
• 8. PlanetMath: Algebraic Geometry
algebraic geometry is the study of algebraic objects using geometrical tools. By algebraic objects, we mean objects such as the collection of solutions to a
http://planetmath.org/encyclopedia/AlgebraicGeometry.html
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##### Introduction
Algebraic geometry is the study of algebraic objects using geometrical tools. By algebraic objects, we mean objects such as the collection of solutions to a list of polynomial equations in some ring . Of course, if the ring is the complex numbers , we can apply the highly succesful theories of complex analysis and complex manifolds to address the problems; many powerful tools are available; de Rham cohomology singular homology Hodge theory spectral sequences and many others. We also have at our disposal all the tools of real differential geometry partitions of unity curvature ... tangent spaces , as well as all the tools of point-set topology . However, if one wishes to use a different ring, perhaps the

9. Mathematical Sciences Research Institute - Algebraic Geometry
algebraic geometry has long been a central subject, with deep and substantial connections to almost every aspect of modern mathematics.
http://www.msri.org/calendar/programs/ProgramInfo/251/show_program
SITE MAP SEARCH SHORTCUT:
Choose a Destination... Calendar Programs Workshops Summer Grad Workshops Seminars Events/Announcements Application Materials Visa Information Propose a Program Propose a Workshop Policy on Diversity MSRI Alumni Archimedes Society Why Give to MSRI Ways to Give to MSRI Donate to MSRI Planned Gifts FAQ Staff Member Directory Contact Us Directions For Visitors Pictures Library Computing SGP Streaming Video / VMath MSRI in the Media Emissary Newsletter Outlook Newsletter Subscribe to Newsletters Books, Preprints, etc. Federal Support Corporate Affiliates Sponsoring Publishers Foundation Support Academic Sponsors
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##### Algebraic Geometry
January 12, 2009 to May 22, 2009 Mathematical Sciences Research Institute, Berkeley, CA. Organized By: William Fulton, Joe Harris, Brendan Hassett, János Kollár, Sándor Kovács, Robert Lazarsfeld, Ravi Vakil Algebraic geometry has long been a central subject, with deep and substantial connections to almost every aspect of modern mathematics. There are numerous different approaches to the field, utilizing widely varying technical tools: Commutative algebra, complex analysis, sheaf theory, cohomological methods, and combinatorial techniques all play important roles.
This multiplicity of techniques and interconnections creates a conundrum for the student of algebraic geometry: How can I learn all the specific tools of the trade, while at the same time developing an overall sense for what guides the subject? Even established researchers can feel powerful centrifugal forces pulling at the field, as it spawns new specialties with astonishing regularity.

10. 14: Algebraic Geometry
algebraic geometry combines the algebraic with the geometric for the benefit of both. Thus the recent proof of Fermat s Last Theorem ostensibly a
http://www.math.niu.edu/~rusin/known-math/index/14-XX.html
Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields
POINTERS: Texts Software Web links Selected topics here
##### Introduction
Algebraic geometry combines the algebraic with the geometric for the benefit of both. Thus the recent proof of "Fermat's Last Theorem" ostensibly a statement in number theory was proved with geometric tools. Conversely, the geometry of sets defined by equations is studied using quite sophisticated algebraic machinery. This is an enticing area but the important topics are quite deep. This area includes elliptic curves.
##### Applications and related fields
• geometry (in particular for conics and curves),
• algebra (since algebraic geometry is commutative ring theory...)
• number theory (especially for Diophantine analysis).
##### Subfields
• Foundations
• Cycles and subschemes
• Families, fibrations
• Birational geometry [Mappings and correspondences]
• 14H: Curves
• 14J: Surfaces and higher-dimensional varieties . For analytic theory, see 32JXX

11. ALGEBRAIC GEOMETRY
This is a graduatelevel text on algebraic geometry that provides a quick and fully self-contained development of the fundamentals, including all
http://www.worldscibooks.com/mathematics/3873.html
 Home Browse by Subject Bestsellers New Titles ... Browse all Subjects Search Bookshop New Titles Editor's Choice Bestsellers Book Series ... Join Our Mailing List ALGEBRAIC GEOMETRY by Daniel Bump (Stanford University) This is a graduate-level text on algebraic geometry that provides a quick and fully self-contained development of the fundamentals, including all commutative algebra which is used. A taste of the deeper theory is given: some topics, such as local algebra and ramification theory, are treated in depth. The book culminates with a selection of topics from the theory of algebraic curves, including the RiemannRoch theorem, elliptic curves, the zeta function of a curve over a finite field, and the Riemann hypothesis for elliptic curves. Contents: Affine Algebraic Sets and Varieties The Extension Theorem Maps of Affine Varieties Dimensions and Products Local Algebra Properties of Affine Varieties Varieties Complete Nonsingular Curves Ramification Completions Differentials and Residues The RiemannRoch Theorem Elliptic Curves and Abelian Varieties The Zeta Function of a Curve Readership: Graduate students in mathematics.

12. The Math Forum - Math Library - Algebraic Geom.
A short article designed to provide an introduction to algebraic geometry, which combines the algebraic with the geometric for the benefit of both.
http://mathforum.org/library/topics/algebraic_g/
 Browse and Search the Library Home Math Topics Geometry : Algebraic Geom. Library Home Search Full Table of Contents Suggest a Link ... Library Help Selected Sites (see also All Sites in this category Algebraic Geometry - Dave Rusin; The Mathematical Atlas A short article designed to provide an introduction to algebraic geometry, which combines the algebraic with the geometric for the benefit of both. Thus the recent proof of "Fermat's Last Theorem" - ostensibly a statement in number theory - was proved with geometric tools. Conversely, the geometry of sets defined by equations is studied using quite sophisticated algebraic machinery. This is an enticing area but the important topics are quite deep. This area includes elliptic curves. Applications and related fields and subfields; textbooks, reference works, and tutorials; software and tables; other web sites with this focus. more>> The Algebraic Geometry Notebooks - Aksel Sogstad For non-experts. Contents: What is algebraic geometry?; What can algebraic geometry be used for?; Classification of curves in the Weierstrass familiy; Bezout's Theorem; Varieties, orbits and orbit-spaces; Automatic theorem proving using Singular. more>> All Sites - 100 items found, showing 1 to 50

13. Mathematical Sciences Research Institute - Algebraic Geometry
In Spring 2009, MSRI will hold a special jumbo halfyear program on algebraic geometry. Jumbo in this context means that this algebraic geometry program
http://www.math.washington.edu/~msri2009/
##### MSRI PROGRAM in ALGEBRAIC GEOMETRY in 2009
In Spring 2009, MSRI will hold a special jumbo half-year program on Algebraic Geometry. "Jumbo" in this context means that this Algebraic Geometry program will be the only program at MSRI during that time as opposed to the usual sharing between two different programs. It will be twice the size of usual programs. The application deadlines have passed. Research Memberships and Postdoctoral Fellowships are being announced by MSRI.
##### Current Workshop Schedule
Connections: Algebraic Geometry and Related Fields January 22, 2009 to January 24, 2009 Organized By: Angela Gibney (U. Pennsylvania), Brendan Hassett (Rice U.), Sándor Kovács (U. Washington), Diane Maclagan (Warwick U.), Jessica Sidman (Mt. Holyoke), and Ravi Vakil (Stanford U.) Classical Algebraic Geometry January 26, 2009 to January 30, 2009 Modern Moduli Theory February 23, 2009 to February 27, 2009 Organized By: Izzet Coskun (U. Illinois - Chicago), Sheldon Katz (U. Illinois), Alina Marian (U. Illinois - Chicago), Rahul Pandharipande (Princeton U.), Richard Thomas (Imperial College), Hsian-Hua Tseng (U. Wisconsin), Ravi Vakil (Stanford U.) Combinatorial, Enumerative and Toric Geometry

14. Journal Of Algebraic Geometry Online
Tables of contents, abstracts and references from vol.11 (2002). Full text to subscribers (TeX, DVI, PDF, PS).
http://www.ams.org/jag/
 Journals Home Search Author Info Subscribe ... All issues Authors Author Packages Initial Submissions Where to send files for accepted papers Manuscript tracking About Editorial Board Subscription Information Subscribe from the AMS Bookstore License agreement The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology. This journal, published quarterly with articles posted individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website. Subscriptions: Subscriptions to this journal can be obtained through the AMS Bookstore . Subscriptions and orders can also be addressed to the American Mathematical Society, P.O. Box 845904, Boston, MA 02284-5904. All orders must be accompanied by payment. A signed license agreement is required prior to electronic access activation.

15. Introduction To Algebraic Geometry
The description in the course guide Introduces the basic notions and techniques of modern algebraic geometry. Algebraic sets, Hilbert s Nullstellensatz
http://math.stanford.edu/~vakil/725/course.html
##### 18.725: Introduction to Algebraic Geometry
The description in the course guide: "Introduces the basic notions and techniques of modern algebraic geometry. Algebraic sets, Hilbert's Nullstellensatz and varieties over algebraically closed fields. We relate varieties over the complex numbers to complex analytic manifolds. For varieties of dimension one (i.e. curves) we discuss the genus, divisors, linear series, line bundles and the Riemann-Roch theorem." Johan de Jong will be teaching the follow-up course in the spring. The class is in Here are various notes in dvi, ps, and pdf formats. (If you would like tar archives, just let me know; due to limited diskspace, I had to remove them from this page.) I've added all the corrections I've found (or been told about) on Dec. 30, 1999, so they are in as final form as they'll ever be.
• On the first day (Sept. 9), I gave out two handouts, one with information about the course ( dvi ps , or pdf ), and one with fun problems in algebraic geometry to pique your interest ( dvi ps , or pdf
Here's a rather detailed summary of the first lecture ( dvi ps , or pdf
• On the second day (Sept. 14), I gave out the first problem set (

16. Algebraic Geometry
algebraic geometry is a subject with historical roots in analytic geometry. At its most naive level it is concerned with the geometry of the solutions of a
http://www.math.tamu.edu/~Peter.Stiller/agpage.html
##### Research Interests:

17. Seattle'05
Three oneweek sessions Interactions with physics; Classical geometry; Arithmetic geometry. University of Washington, Seattle, WA, USA; 25 July 12
http://www.math.princeton.edu/~rahulp/seattle05.html
 Summer Institute in Algebraic Geometry July 25 - August 12, 2005 Program: Plenary / Full AMS information brochure Participant list Graduate student workshop The American Mathematical Society, the Clay Mathematics Institute, and the National Science Foundation will sponsor a three week Summer Institute in Algebraic Geometry at the University of Washington, Seattle from July 25 to August 12, 2005. The goals of the Institute are to review the major achievements of the past decade and to look forward to future developments. The focus will be structured by week: Interactions with physics Classical geometry Arithmetic geometry Plenary lecture series aimed at broad audiences will be scheduled in the morning. More specialized seminar sessions will take place in the afternoon. Financial support, as always, will be limited. Participants are encouraged to seek travel funds from their home institutions or funding agencies. The organizing committee

18. IMA Thematic Year On Applications Of Algebraic Geometry, September 2006 - June 2
algebraic geometry has a long and distinguished presence in the history of mathematics that produced both powerful and elegant theorems.
http://www.ima.umn.edu/2006-2007/
IMA home Contact IMA What's inside
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##### Applications of Algebraic Geometry
September 2006 - June 2007 Poster Request Form JPEG low resolution medium resolution high resolution PDF full resolution (128 MB) 2006-2007 Annual Report: pdf Questions? Contact us at domain='ima.umn.edu'; document.write('staff@'); document.write('ima.umn.edu'); staff(at)ima.umn.edu Long Term Visitors Events Participants 2006-2007 Postdoctoral Fellows Algebraic Geometry and Applications Seminar ... Weekly Tutorial: Real Algebraic Geometry Quick Links to Events Tutorial: Algebraic geometric methods in engineering Workshop: Algorithms in algebraic geometry "Hot topics" workshop: Negative index materials Workshop: Software for algebraic geometry Public lecture: Margaret H. Wright How hard can it be? Special workshop: Blackwell-Tapia conference Math is Cool! and Who Wants to Be a Mathematician? "Hot topics" workshop: The evolution of mathematical communication in the age of digital libraries Tutorial: Algebraic algorithms in optimization Workshop: Optimization and control Public Lecture: Christopher J. Budd

19. School Of Mathematics - New Connections Of Representation Theory To Algebraic Ge
During the 200708 academic year, Roman Bezrukavnikov of MIT will lead a special program on algebraic geometry and physics in representation theory.
http://www.math.ias.edu/pages/activities/special-programs/new-connections-of-rep
• Home Activities
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##### New Connections of Representation Theory to Algebraic Geometry and Physics
During the 2007-08 academic year, Roman Bezrukavnikov of MIT will lead a special program on algebraic geometry and physics in representation theory. The focus of the year will be on related recent developments in representation theory, algebraic geometry and physics.
The first conference, "Gauge Theory and Representation Theory", will be held during the week of November 26-30, 2007.
The second conference, "Algebro-Geometric Derived Categories and Applications", will be held during term II from March 10-14, 2008.
"For both conferences there will be financial support available for the participation of students, minorities, women and postdoctoral scholars.Â  Closer to the date of the conference, please refer to the conference website for information about how to apply for financial support."
In the first term the emphasis will be on mathematics connected to quantum field theory, in particular the new differential geometric approach to the geometric Langlands program. A part of the second term will be devoted to absorbing the emerging new homotopy foundations of algebraic geometry, with a view towards applications.
One common feature of recent trends is âcategorificationâ, often synonymous with âgeometrizationâ. Categorification stands for the passage from a traditional mathematical object to its higher categorical analogue, and, more loosely, for the emphasis on categories instead of particular objects. The categories involved are typically of geometric nature (categories of sheaves of certain kind) and are constructed in a homological framework, i.e., they are triangulated categories, or refinements of these. Examples in representation theory include geometric Langlands duality (a categorification of the theory of automorphic forms); character sheaves (a categorification of representation theory of finite Chevalley groups); localization techniques for modular representations; Nakajimaâs geometric construction of Kac-Moody Lie algebra representations etc. However, there are many examples in other fields which are relevant for representation theory: categories of

20. MIT OpenCourseWare | Mathematics | 18.725 Algebraic Geometry, Fall 2003 | Lectur
Back to algebraic geometry. 6, Review of things not covered enough (Topics Fibers, Morphisms of Sheaves) (PDF) Back to Work Morphisms Varieties
http://ocw.mit.edu/OcwWeb/Mathematics/18-725Fall2003/LectureNotes/
• Home Courses Donate About OCW ... Mathematics Algebraic Geometry
##### Lecture Notes
The lecture notes are courtesy Moses Liskov, a student in the class. Used with permission.
LEC # TOPICS Introduction ( PDF
First Goal Recap of Last Time ( PDF
Irreducible Components
Projective Space Projective Space (cont.) ( PDF Projective Space (cont.) ( PDF
Topological Diversion
Sheaves Presheaves ( PDF
Back to Algebraic Geometry Review of things not covered enough (Topics: Fibers, Morphisms of Sheaves) ( PDF
Back to Work
Morphisms
Varieties Homework Review ( PDF Back to Varieties Projective Varieties ( PDF A Review on Projective Varieties ( PDF Product of Varieties Applications ( PDF Recap on the Applications ( PDF Next Definition of Variety Dimension Homework 5 Problem ( PDF Dimension Future Applications Back to Dimension ( PDF Krull Dimension Fibers of Morphisms ( PDF Fiber Products ( PDF Back to the Theorem Complete Varities Completeness ( PDF Complex Topology Chow's Lemma ( PDF Analytic Spaces Curves ( PDF What is Next? (

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