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