In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field k is the zero-locus in the affine space kn of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set. A Zariski open subvariety of an affine variety is called a quasi-affine variety. Some texts do not require a prime ideal, and call irreducible an algebraic variety defined by a prime ideal. This article refers to zero-loci of not necessarily prime ideals as affine algebraic sets. In some contexts, it is useful to distinguish the field k in which the coefficients are considered, from the algebraically closed field K (containing k) over which the zero-locus is considered (that is, the points of the affine variety are in Kn). In this case, the variety is said defined over k, and the points of the variety that belong to kn are said k-rational or rational over k. In the common case where k is the field of real numbers, a k-rational point is called a real point. When the field k is not specified, a rational point is a point that is rational over the rational numbers. For example, Fermat's Last Theorem asserts that the affine algebraic variety (it is a curve) defined by xn + yn − 1 = 0 has no rational points for any integer n greater than two. (Wikipedia).

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

Affine and mod-affine varieties in arithmetic geometry. - Charles - Workshop 2 - CEB T2 2019

François Charles (Université Paris-Sud) / 24.06.2019 Affine and mod-affine varieties in arithmetic geometry. We will explain how studying arithmetic versions of affine schemes and their bira- tional modifications leads to a generalization to arbitrary schemes of both Fekete’s theorem on

From playlist 2019 - T2 - Reinventing rational points

algebraic geometry 5 Affine space and the Zariski topology

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the definition of affine space and its Zariski topology.

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

Some results on affine Deligne–Lusztig varieties – Xuhua He – ICM2018

Lie Theory and Generalizations Invited Lecture 7.8 Some results on affine Deligne–Lusztig varieties Xuhua He Abstract: The study of affine Deligne–Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne–Lusztig varieties are purely Lie-theoretic i

From playlist Lie Theory and Generalizations

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 18 Products of varieties

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers products of affine and projective varieties and the Segre embedding.

From playlist Algebraic geometry I: Varieties

Novel Algebraic Operations for Affine Geometry | Algebraic Calculus One | Wild Egg

We introduce some novel conventions to help us set up the foundations of affine geometry. We learn about differences of points, sums of points and vectors, affine combinations and vector proportions. And then use these to state a number of important results from affine geometry, including

From playlist Algebraic Calculus One from Wild Egg

Calculus and affine geometry of the magical parabola | Algebraic Calc and dCB curves 3 | Wild Egg

Algebraic Calculus naturally lives in affine geometry, not Euclidean geometry. Affine geometry is the geometry of parallelism, or (almost the same thing) --- the geometry of pure linear algebra. The parabola is characterized projectively in this geometry as the unique conic which is tangen

From playlist Algebraic Calculus One Info

Laura Rider: Modular Perverse Sheaves on the affine Flag Variety

There are two categorical realizations of the affine Hecke algebra: constructible sheaves on the affine flag variety and coherent sheaves on the Langlands dual Steinberg variety. A fundamental problem in geometric representation theory is to relate these two categories by a category equiva

From playlist Workshop: Monoidal and 2-categories in representation theory and categorification

algebraic geometry 27 The twisted cubic

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It describes two examples: the twisted cubic is isomorphic to a projective line, and the affine plane without the origin is not isomorphic to any affine algebraic set.

From playlist Algebraic geometry I: Varieties

Robert Cass: Perverse mod p sheaves on the affine Grassmannian

28 September 2021 Abstract: The geometric Satake equivalence relates representations of a reductive group to perverse sheaves on an affine Grassmannian. Depending on the intended application, there are several versions of this equivalence for different sheaf theories and versions of the a

From playlist Representation theory's hidden motives (SMRI & Uni of Münster)

Anthony Henderson: Hilbert Schemes Lecture 1

SMRI Seminar Series: 'Hilbert Schemes' Lecture 1 Introduction Anthony Henderson (University of Sydney) This series of lectures aims to present parts of Nakajima’s book `Lectures on Hilbert schemes of points on surfaces’ in a way that is accessible to PhD students interested in representa

From playlist SMRI Course: Hilbert Schemes

Modular Perverse Sheaves on the affine Flag Variety - Laura Rider

Virtual Workshop on Recent Developments in Geometric Representation Theory Topic: Modular Perverse Sheaves on the affine Flag Variety Speaker: Laura Rider Affiliation: University of Georgia Date: November 16, 2020 For more video please visit http://video.ias.edu

From playlist Virtual Workshop on Recent Developments in Geometric Representation Theory

A Gentle Approach to Crystalline Cohomology - Jacob Lurie

Members’ Colloquium Topic: A Gentle Approach to Crystalline Cohomology Speaker: Jacob Lurie Affiliation: Professor, School of Mathematics Date: February 28, 2022 Let X be a smooth affine algebraic variety over the field C of complex numbers (that is, a smooth submanifold of C^n which can

From playlist Mathematics

Schemes 21: Separated morphisms

This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne.. We define separated and quasi-separated schemes and morphisms, give a few examples, and show that if a scheme has a separated morphism to an affine scheme the

From playlist Algebraic geometry II: Schemes

Tropical Geometry - Lecture 3 - Fields and Varieties | Bernd Sturmfels

Twelve lectures on Tropical Geometry by Bernd Sturmfels (Max Planck Institute for Mathematics in the Sciences | Leipzig, Germany) We recommend supplementing these lectures by reading the book "Introduction to Tropical Geometry" (Maclagan, Sturmfels - 2015 - American Mathematical Society)

From playlist Twelve Lectures on Tropical Geometry by Bernd Sturmfels

BAG2.3. Affine Pieces of Projective Toric Varieties

Basic Algebraic Geometry: This part has three goals: formalizing some notions used in the previous parts; noting a result about tori; and begin study of affine pieces of projective toric varieties.

From playlist Basic Algebraic Geometry