Algebraic varieties | Algebraic geometry
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is a defining feature of algebraic geometry. Many algebraic varieties are manifolds, but an algebraic variety may have singular points while a manifold cannot. Algebraic varieties can be characterized by their dimension. Algebraic varieties of dimension one are called algebraic curves and algebraic varieties of dimension two are called algebraic surfaces. In the context of modern scheme theory, an algebraic variety over a field is an integral (irreducible and reduced) scheme over that field whose structure morphism is separated and of finite type. (Wikipedia).
algebraic geometry 25 Morphisms of varieties
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the definition of a morphism of varieties and compares algebraic varieties with other types of locally ringed spaces.
From playlist Algebraic geometry I: Varieties
Algebraic geometry 44: Survey of curves
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It gives an informal survey of complex curves of small genus.
From playlist Algebraic geometry I: Varieties
algebraic geometry 15 Projective space
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It introduces projective space and describes the synthetic and analytic approaches to projective geometry
From playlist Algebraic geometry I: Varieties
algebraic geometry 22 Toric varieties
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It describes toric varieties as examples of abstract varieties. For more about these see the book "Introduction to toric varieties" by Fulton.
From playlist Algebraic geometry I: Varieties
algebraic geometry 26 Affine algebraic sets and commutative rings
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the relation between morphisms of affine algebraic sets and homomorphisms of commutative rings. As examples it describes some homomorphisms of commutative rings
From playlist Algebraic geometry I: Varieties
algebraic geometry 24 Regular functions
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers regular functions on affine and quasiprojective varieties.
From playlist Algebraic geometry I: Varieties
algebraic geometry 23 Categories
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It gives a quick review of category theory as background for the definition of morphisms of algebraic varieties.
From playlist Algebraic geometry I: Varieties
algebraic geometry 29 Automorphisms of space
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It describes the automorphisms of affine and projective space, and gives a brief discussion of the Jacobian conjecture.
From playlist Algebraic geometry I: Varieties
algebraic geometry 17 Affine and projective varieties
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the relation between affine and projective varieties, with some examples such as a cubic curve and the twisted cubic.
From playlist Algebraic geometry I: Varieties
Title: Differential Varieties with Only Algebraic Images
From playlist Fall 2014
Ariyan Javanpeykar: Arithmetic and algebraic hyperbolicity
Abstract: The Green-Griffiths-Lang-Vojta conjectures relate the hyperbolicity of an algebraic variety to the finiteness of sets of “rational points”. For instance, it suggests a striking answer to the fundamental question “Why do some polynomial equations with integer coefficients have onl
From playlist Algebraic and Complex Geometry
Title: The Dixmier-Moeglin Problem for D-Varieties May 2016 Kolchin Seminar Workshop
From playlist May 2016 Kolchin Seminar Workshop
Finiteness theorems for Kolchin's constrained cohomology
By Anand Pillay, University of Notre Dame Finiteness theorems for Kolchin's constrained cohomology Kolchin Seminar, CUNY Graduate Center, October 4, 2019
From playlist Fall 2019 Kolchin Seminar in Differential Algebra
Nonlinear algebra, Lecture 2: "Algebraic Varieties", by Mateusz Michałek
This is the second lecture in the IMPRS Ringvorlesung, the advanced graduate course at the Max Planck Institute for Mathematics in the Sciences. In this lecture, Mateusz Michalek describes the main characters in algebraic geometry: algebraic varieties.
From playlist IMPRS Ringvorlesung - Introduction to Nonlinear Algebra
Bernd Sturmfels (8/28/18): Learning algebraic varieties from samples
We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of
From playlist AATRN 2018
Functional transcendence and arithmetic applications – Jacob Tsimerman – ICM2018
Number Theory Invited Lecture 3.13 Functional transcendence and arithmetic applications Jacob Tsimerman Abstract: We survey recent results in functional transcendence theory, and give arithmetic applications to the André–Oort conjecture and other unlikely-intersection problems. © Int
From playlist Number Theory
Finite or infinite? One key to algebraic cycles - Burt Totaro
Burt Totaro University of California, Los Angeles; Member, School of Mathematics February 2, 2015 Algebraic cycles are linear combinations of algebraic subvarieties of an algebraic variety. We want to know whether all algebraic subvarieties can be built from finitely many, in a suitable s
From playlist Mathematics
Nonlinear algebra, Lecture 7: "Toric Varieties", by Mateusz Michalek
This is the seventh lecture in the IMPRS Ringvorlesung, the advanced graduate course at the Max Planck Institute for Mathematics in the Sciences.
From playlist IMPRS Ringvorlesung - Introduction to Nonlinear Algebra
Representation theory of W-algebras and Higgs branch conjecture – Tomoyuki Arakawa – ICM2018
Lie Theory and Generalizations Invited Lecture 7.2 Representation theory of W-algebras and Higgs branch conjecture Tomoyuki Arakawa Abstract: We survey a number of results regarding the representation theory of W-algebras and their connection with the resent development of the four dimen
From playlist Lie Theory and Generalizations
algebraic geometry 31 Rational maps
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the definition of rational functions and rational maps, and gives an example of a cubic curve that is not birational to the affine line.
From playlist Algebraic geometry I: Varieties