Ring theory | Projective geometry | Algebraic geometry
In mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring A with 1, the projective line P(A) over A consists of points identified by projective coordinates. Let U be the group of units of A; pairs (a, b) and (c, d) from A × A are related when there is a u in U such that ua = c and ub = d. This relation is an equivalence relation. A typical equivalence class is written U[a, b]. P(A) = { U[a, b] : aA + bA = A }, that is, U[a, b] is in the projective line if the ideal generated by a and b is all of A. The projective line P(A) is equipped with a group of homographies. The homographies are expressed through use of the matrix ring over A and its group of units V as follows: If c is in Z(U), the center of U, then the group action of matrix on P(A) is the same as the action of the identity matrix. Such matrices represent a normal subgroup N of V. The homographies of P(A) correspond to elements of the quotient group V / N . P(A) is considered an extension of the ring A since it contains a copy of A due to the embedding E : a → U[a, 1]. The multiplicative inverse mapping u → 1/u, ordinarily restricted to the group of units U of A, is expressed by a homography on P(A): Furthermore, for u,v ∈ U, the mapping a → uav can be extended to a homography: Since u is arbitrary, it may be substituted for u−1.Homographies on P(A) are called linear-fractional transformations since (Wikipedia).
This lecture is part of an online course on rings and modules. We define projective modules, and give severalexamples of them, including the Moebius band, a non-principal ideal, and the tangent bundle of the sphere. For the other lectures in the course see https://www.youtube.com/playli
From playlist Rings and modules
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
Introduction to Projective Geometry (Part 1)
The first video in a series on projective geometry. We discuss the motivation for studying projective planes, and list the axioms of affine planes.
From playlist Introduction to Projective Geometry
Definition of a Ring and Examples of Rings
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Definition of a Ring and Examples of Rings - Definition of a Ring. - Definition of a commutative ring and a ring with identity. - Examples of Rings include: Z, Q, R, C under regular addition and multiplication The Ring of all n x
From playlist Abstract Algebra
The circle and projective homogeneous coordinates | Universal Hyperbolic Geometry 7a | NJ Wildberger
Universal hyperbolic geometry is based on projective geometry. This video introduces this important subject, which these days is sadly absent from most undergrad/college curriculums. We adopt the 19th century view of a projective space as the space of one-dimensional subspaces of an affine
From playlist Universal Hyperbolic Geometry
BAG2.2. Projective Toric Varieties - Part 2
Basic Algebraic Geometry: Continuing from the previous video, we give several equivalent conditions for when the cone over the projective toric variety X_A is equal to the affine toric variety Y_A.
From playlist Basic Algebraic Geometry
The circle and projective homogeneous coordinates (cont.) | Universal Hyperbolic Geometry 7b
Universal hyperbolic geometry is based on projective geometry. This video introduces this important subject, which these days is sadly absent from most undergrad/college curriculums. We adopt the 19th century view of a projective space as the space of one-dimensional subspaces of an affine
From playlist Universal Hyperbolic Geometry
Elliptic curves: point at infinity in the projective plane
This video depicts point addition and doubling on elliptic curve in simple Weierstrass form in the projective plane depicted using stereographic projection where the point at infinity can actually be seen. Explanation is in the accompanying article https://trustica.cz/2018/04/05/elliptic-
From playlist Elliptic Curves - Number Theory and Applications
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. We give two examples of gluing affine schemes to get non-affine schemes: the line with two origins and the projective line. We calculate the regular functions
From playlist Algebraic geometry II: Schemes
Schemes 41: Morphisms to projective space
This lecture is part of an online course on algebraic geometry based on chapter II of "algebraic geometry" by Hartshorne. We discuss morphisms of a scheme to projective space, showing that they correspond to a line bundle with a set of sections generating it.
From playlist Algebraic geometry II: Schemes
Duality in Algebraic Geometry by Suresh Nayak
PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics
From playlist Dualities in Topology and Algebra (Online)
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. We describe the construction of Proj S, a scheme associated to a graded ring S, that generalizes the construction of a projective variety.
From playlist Algebraic geometry II: Schemes
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
BAG2.1. Projective Toric Varieties - Part 1
Basic Algebraic Geometry: We define complex projective space, projective varieties, and projective toric varieties. For PTVs, we identify the character lattice and lattice of one-parameter subgroups.
From playlist Basic Algebraic Geometry
Elliptic Curves - Lecture 4a - Varieties, function fields, dimension
This video is part of a graduate course on elliptic curves that I taught at UConn in Spring 2021. The course is an introduction to the theory of elliptic curves. More information about the course can be found at the course website: https://alozano.clas.uconn.edu/math5020-elliptic-curves/
From playlist An Introduction to the Arithmetic of Elliptic Curves
Schemes 23: Valuations and separation
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne.. We state a condition for morphisms of schemes to be separated in therms of discrete valuation rings, and apply this to the line with two origins and the proje
From playlist Algebraic geometry II: Schemes
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
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
Algebraic geometry 49: Hilbert polynomials
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It gives a review of the Hilbert polynomial of a graded module over a graded ring, and classifies integer-valued polynomials.
From playlist Algebraic geometry I: Varieties
Marta Pieropan, The split torsor method for Manin’s conjecture
See https://tinyurl.com/y98dn349 for an updated version of the slides with minor corrections. VaNTAGe seminar 20 April 2021
From playlist Manin conjectures and rational points