Algebraic Geometry

Mathematics Geometry

Algebra Linear Algebra Homological Algebra Commutative Algebra Category Theory Algebraic Topology

Algebraic Curve Algebraic Variety Affine Space Projective Space

Riemann Surface Scheme Stack Sheaf Gerbe

Tropical Geometry


Algebraic geometry is a branch of mathematics that studies geometric objects defined by polynomial equations and their properties. It involves the use of algebraic tools, such as rings, fields, and modules, to investigate the geometric properties of algebraic varieties, which are sets of solutions to polynomial equations. Algebraic geometry has applications in many areas of mathematics, including number theory, topology, and differential geometry, as well as in physics and engineering. Its techniques have been used to solve long-standing problems, such as Fermat's Last Theorem, and it continues to be an active area of research with many open problems and conjectures.


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 x^2+y^2=1 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 algebraic number theory. For instance, Deligne used it to prove a variant of the Riemann hypothesis. Also, Andrew Wiles' proof of Fermat's last theorem used the tools developed in algebraic geometry.

In the latter part of the twentieth century, researchers have tried to extend the relationship between algebra and geometry to arbitrary noncommutative rings. The study of geometries associated to noncommutative rings is called noncommutative geometry.


Algebraic Varieties: Algebraic varieties are the central objects of study in algebraic geometry. They are defined as the solution sets of systems of polynomial equations. For example, the circle in the plane can be defined by the equation x^2 + y^2 = 1, making it an algebraic variety.

Projective Geometry: Projective geometry is a geometric setting where parallel lines meet at a point at infinity. Projective spaces generalize Euclidean spaces and provide a natural setting for studying algebraic varieties.

Schemes: Schemes are a generalization of algebraic varieties that allows for a more flexible and powerful approach to studying algebraic geometry. They are defined using the language of commutative algebra and are fundamental in modern algebraic geometry.

Sheaf Theory: Sheaf theory is a tool used in algebraic geometry to study the local properties of algebraic varieties. It provides a way to understand how functions defined on open sets of a variety glue together to define global functions.

Coherent Sheaves: Coherent sheaves are a special class of sheaves that play a crucial role in algebraic geometry. They are used to study vector bundles, which are geometric objects that locally look like products of a variety with a vector space.

Intersection Theory: Intersection theory studies the intersections of algebraic varieties in a way that respects their multiplicities. It provides a way to count the number of common points of two varieties and has applications in enumerative geometry.

Moduli Spaces: Moduli spaces parameterize families of algebraic varieties. They are used to study the geometric properties of varieties and to classify them up to certain equivalence relations.


Riemann-Roch Theorem: This theorem relates the topological properties of a compact Riemann surface to its algebraic properties, providing a powerful tool for computing the dimensions of spaces of meromorphic functions.

Hodge Theory: Hodge theory studies the relationship between the algebraic and topological properties of complex manifolds. It decomposes the cohomology groups of a complex manifold into harmonic forms, providing insights into the geometric structure of the manifold.

Intersection Theory: Intersection theory studies the intersections of subvarieties in algebraic geometry and relates them to the cohomology of the ambient space. It has applications in enumerative geometry and the study of algebraic cycles.

Toric Geometry: Toric geometry studies algebraic varieties that arise from the action of a torus on an affine space. It provides a combinatorial description of these varieties, allowing for the study of their geometric properties using combinatorial techniques.

Mirror Symmetry: Mirror symmetry is a conjectural duality between certain pairs of Calabi-Yau manifolds, which are special kinds of algebraic varieties. It has deep connections to string theory and has led to many new insights in algebraic geometry.

Derived Categories: Derived categories are a way to extend the notion of homological algebra to a wider class of objects. They play a crucial role in modern algebraic geometry, particularly in the study of moduli spaces and deformation theory.

Stacks: Stacks are a generalization of schemes that allow for more flexible and functorial constructions. They are used to study moduli problems and have applications in algebraic geometry and mathematical physics.


http://mathonline.wikidot.com/geometry


https://www.jmilne.org/math/index.html


Tutorials

https://rigtriv.wordpress.com/ag-from-the-beginning/

https://web.archive.org/web/20161022062721/alpha.math.uga.edu/~roy/introAG.pdf A Naive Introduction to Algebraic Geometry - Roy Smith

Resources

http://alpha.math.uga.edu/~pete/integral2015.pdf Commutative Algebra - Pete L. Clark

https://math.uchicago.edu/~amathew/CRing.pdf The CRing Project

http://stacks.math.columbia.edu The Stacks Project

http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf The Rising Sea: Foundations of Algebraic Geometry - Ravi Vakil

http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf Algebraic Geometry [Lecture Notes] - Andreas Gathman

http://neo-classical-physics.info/uploads/3/0/6/5/3065888/van_der_waerden_-_algebraic_geometry.pdf

http://www2.math.uu.se/~khf/Snok.pdf Algebraic Geometry - Karl-Heinz Fieseler and Ludger Kaup

http://web.archive.org/web/20150616043837/http://www.risc.jku.at/research/alggeom/seminar/files/tutorialAG.pdf TUTORIAL ON ALGEBRAIC GEOMETRY - JANKA PÍLNIKOVÁ

http://homepages.math.uic.edu/~marker/math494/eag.pdf Elementary Algebraic Geometry - David Marker

http://www.silicovore.com/agathos/contents.html AGATHOS - Algebraic Geometry: A Total Hypertext Online System - Kevin R. Coombes

https://www.uni-frankfurt.de/50581209/einfalggeom11.pdf Einführung in die Algebraische Geometrie - Annette Werner

https://courses.cs.washington.edu/courses/cse590b/13au/ Algebraic Geometry for Computer Graphics - James F. Blinn


Affine Maps, Euclidean Motions and Quadrics - Agustí Reventós Tarrida

Rational points on elliptic curves - Joseph H. Silverman, John Tate

Elementary algebraic geometry - Klaus Hulek

Plane Algebraic Curves - Gerd Fischer

Algebraic Geometry and Commutative Algebra - Siegfried Bosch

Algebraic Geometry: An Introduction - Daniel Perrin

Algebraic Geometry - Ulrich Görtz

Lectures on Algebraic Geometry - Günter Harder

Daniel Coray - Notes on Geometry and Arithmetic


Jean-Pierre Serre - Galois Cohomology

Thomas Dedieu, Flaminio Flamini, Claudio Fontanari, Concettina Galati, Rita Pardini - The Art of Doing Algebraic Geometry


https://kerodon.net/ an online resource for homotopy-coherent mathematics - Jacob Lurie


https://www.encyclopediaofmath.org/index.php/Analytic_surface_%28in_algebraic_geometry%29

https://encyclopediaofmath.org/wiki/Cubic_hypersurface

https://encyclopediaofmath.org/wiki/Riemann_surface

https://en.wikipedia.org/wiki/Invariant_theory

https://encyclopediaofmath.org/wiki/Lefschetz_formula

https://en.wikipedia.org/wiki/Nef_line_bundle

https://encyclopediaofmath.org/index.php/Cubic_hypersurface

https://encyclopediaofmath.org/wiki/Mori_theory_of_extremal_rays


https://ahilado.wordpress.com/


https://homepages.warwick.ac.uk/~maseap/


https://homepages.warwick.ac.uk/~maseap/arith/notes/


https://nsalter.science.nd.edu/expository-notes/


https://web.ma.utexas.edu/users/benzvi/Langlands.html


Miscellaneous

https://web.archive.org/web/20110721024709/http://perso.univ-rennes1.fr/antoine.chambert-loir/publications/

https://mathoverflow.net/questions/324826/borels-presentation-for-the-cohomology-of-a-flag-variety

https://mathoverflow.net/questions/31650/modern-algebraic-geometry-vs-classical-algebraic-geometry/31730

https://mathoverflow.net/questions/303038/spectral-algebraic-geometry-vs-derived-algebraic-geometry-in-positive-characteri/303190

https://math.stackexchange.com/questions/4230232/torsion-subsheaf-of-coherent-sheaf-on-locally-noetherian-scheme

https://arxiv.org/PS_cache/math/pdf/9911/9911199v1.pdf Algebraic stacks - Tomás L. Gómez

https://people.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Notes on Crystals and Algebraic D-Modules

https://arxiv.org/pdf/1204.2734.pdf Geometry of theta divisors: a survey - Samuel Grushevsky and Klaus Hulek

https://arxiv.org/pdf/2301.05542.pdf Tangent categories as a bridge between differential geometry and algebraic geometry -nG.S.H. Cruttwell, Jean-Simon Pacaud Lemay

https://people.brandeis.edu/~igusa/Math202aF14/Math202aLecturesC.pdf

https://ksda.ccny.cuny.edu/PostedPapers/rickksda1107.pdf

https://terrytao.wordpress.com/2011/07/15/pappuss-theorem-and-elliptic-curves/

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