Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Variety Defined as the Closure of VCA

In course

MATH-328: Algebraic geometry I - Curves

Algebraic geometry is the common language for many branches of modern research in mathematics. This course gives an introduction to this field by studying algebraic curves and their intersection theory. ...

Description

This lecture covers the concept of variety defined as the closure of VCA, exploring its properties and applications in algebraic geometry. The instructor discusses the relationship between varieties and K-algebraic homomorphisms, emphasizing the importance of understanding the kernel and image under various transformations. The lecture delves into local rings and their significance in characterizing points within a variety. Through a series of examples and theoretical explanations, the instructor illustrates how to differentiate between different types of varieties and their corresponding structures.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructor

Dimitri Stelio Wyss

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Geometry: Algebraic geometry

Related concepts (32)

Local ring

In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal.

Linux kernel

The Linux kernel is a free and open-source, monolithic, modular, multitasking, Unix-like operating system kernel. It was originally written in 1991 by Linus Torvalds for his i386-based PC, and it was soon adopted as the kernel for the GNU operating system, which was written to be a free (libre) replacement for Unix. Linux is provided under the GNU General Public License version 2 only, but it contains files under other compatible licenses.

Regular local ring

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of generators of m. Then by Krull's principal ideal theorem n ≥ dim A, and A is defined to be regular if n = dim A. The appellation regular is justified by the geometric meaning.

Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined over that field.

Hybrid kernel

A hybrid kernel is an operating system kernel architecture that attempts to combine aspects and benefits of microkernel and monolithic kernel architectures used in operating systems. The traditional kernel categories are monolithic kernels and microkernels (with nanokernels and exokernels seen as more extreme versions of microkernels). The "hybrid" category is controversial, due to the similarity of hybrid kernels and ordinary monolithic kernels; the term has been dismissed by Linus Torvalds as simple marketing.

Related lectures (48)

Varieties with nef anti-canonical: Surjective Albanese

Presents a proof that smooth projective varieties with nef anti-canonical divisor have surjective Albanese morphism.

Quotients de groupe: Un cas particulier

Explores a categorical perspective on group quotients, focusing on a specific case.

Linear Algebra: Subspaces and Transformations

Explores subspaces in linear algebra and transformations, including kernels and images of linear transformations.

Projective Varieties: An Algebraic Study Log in to Mediaspace to watch this video

Covers the study of projective varieties and their relation to compact manifolds.

Regularity and Geometric Meaning Log in to Mediaspace to watch this video