In algebraic geometry, the scheme-theoretic intersection of closed subschemes X, Y of a scheme W is , the fiber product of the closed immersions . It is denoted by . Locally, W is given as for some ring R and X, Y as for some ideals I, J. Thus, locally, the intersection is given as Here, we used (for this identity, see tensor product of modules#Examples.) Example: Let be a projective variety with the homogeneous coordinate ring S/I, where S is a polynomial ring. If is a hypersurface defined by some homogeneous polynomial f in S, then If f is linear (deg = 1), it is called a hyperplane section. See also: Bertini's theorem. Now, a scheme-theoretic intersection may not be a correct intersection, say, from the point of view of intersection theory. For example, let = the affine 4-space and X, Y closed subschemes defined by the ideals and . Since X is the union of two planes, each intersecting with Y at the origin with multiplicity one, by the linearity of intersection multiplicity, we expect X and Y intersect at the origin with multiplicity two. On the other hand, one sees the scheme-theoretic intersection consists of the origin with multiplicity three. That is, a scheme-theoretic multiplicity of an intersection may differ from an intersection-theoretic multiplicity, the latter given by Serre's Tor formula. Solving this disparity is one of the starting points for derived algebraic geometry, which aims to introduce the notion of . (Wikipedia).
Geogebra Tutorial : Union and Intersection of Sets
Union and intersection of sets can be drawing with geogebra. Please see the video to start how drawing union and intersection of sets. more visit https://onwardono.com
From playlist SET
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne.. We define fibered products of schemes, sketch their construction, and give a few examples to illustrate their slightly odd behavior.
From playlist Algebraic geometry II: Schemes
Equivalence Relations Definition and Examples
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Equivalence Relations Definition and Examples. This video starts by defining a relation, reflexive relation, symmetric relation, transitive relation, and then an equivalence relation. Several examples are given.
From playlist Abstract Algebra
Geometric Algebra - Duality and the Cross Product
In this video, we will introduce the concept of duality, involving a multiplication by the pseudoscalar. We will observe the geometric meaning of duality and also see that the cross product and wedge product are dual to one another, which means that the cross product is already contained w
From playlist Geometric Algebra
What is an Intersection? (Set Theory)
What is the intersection of sets? This is another video on set theory in which we discuss the intersection of a set and another set, using the classic example of A intersect B. We do not quite go over a formal definition of intersection of a set in this video, but we come very close! Be su
From playlist Set Theory
This lecture is part of an online course in algebraic geometry giving an introduction to schemes. It is loosely based on chapter II Hartshorne's book "Algebraic geometry". (For chapter 1 see the playlist "Algebraic geometry".) This introductory lecture gives some motivation for schemes and
From playlist Algebraic geometry II: Schemes
Davesh Maulik - Stable Pairs and Gopakumar-Vafa Invariants 1/5
In the first part of the course, I will give an overview of Donaldson-Thomas theory for Calabi-Yau threefold geometries, and its cohomological refinement. In the second part, I will explain a conjectural ansatz (from joint work with Y. Toda) for defining Gopakumar-Vafa invariants via modul
From playlist 2021 IHES Summer School - Enumerative Geometry, Physics and Representation Theory
Andrei Negut: Hilbert schemes of K3 surfaces
Abstract: We give a geometric representation theory proof of a mild version of the Beauville-Voisin Conjecture for Hilbert schemes of K3 surfaces, namely the injectivity of the cycle map restricted to the subring of Chow generated by tautological classes. Although other geometric proofs o
From playlist Algebraic and Complex Geometry
Artan Sheshmani : On the proof of S-duality modularity conjecture on quintic threefolds
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Algebraic and Complex Geometry
Schemes 5: Definition of a scheme
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. We give some historical background, then give the definition of a scheme and some simple examples, and finish by explaining the origin of the word "spectrum".
From playlist Algebraic geometry II: Schemes
Anthony Licata: Hilbert Schemes Lecture 7
SMRI Seminar Series: 'Hilbert Schemes' Lecture 7 Kleinian singularities 2 Anthony Licata (Australian National University) 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 inter
From playlist SMRI Course: Hilbert Schemes
Richard Thomas - Vafa-Witten Invariants of Projective Surfaces 5/5
This course has 4 sections split over 5 lectures. The first section will be the longest, and hopefully useful for the other courses. 1. Sheaves, moduli and virtual cycles 2. Vafa-Witten invariants: stable and semistable cases 3. Techniques for calculation --- virtual degeneracy loci, cose
From playlist 2021 IHES Summer School - Enumerative Geometry, Physics and Representation Theory
The vector cross-product is another form of vector multiplication and results in another vector. In this tutorial I show you a simple way of calculating the cross product of two vectors.
From playlist Introducing linear algebra
Nexus trimester - Omri Weinstein (Courant Institute (NYU)) 2/6
Some Some Information-Theoretic Problems in Theoretical Computer Science - Part II February 04, 2016 Abstract: In this informal talk, I will present and shortly discuss a few long-standing open problems in theoretical computer science (TCS), including Secret-Sharing, Multi-terminal commu
From playlist Nexus Trimester - 2016 - Distributed Computation and Communication Theme
Joshua Ciappara: Hilbert Schemes Lecture 10
SMRI Seminar Series: 'Hilbert Schemes' Lecture 10 Representations of Heisenberg algebras on homology of Hilbert schemes Joshua Ciappara (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 tha
From playlist SMRI Course: Hilbert Schemes
Sites/Coverings Examples part 1
We give the baby examples of sites in our new language.
From playlist Sites, Coverings and Grothendieck Topologies
Intersection and union of sets 2
drawing intersection and union with geogebra. this video can help you to drawing sets.
From playlist Go Geogebra
Nexus Trimester - Salim El Rouayheb (Illinois Institute of Technology)
Secret Sharing in Distributed Storage Systems Salim El Rouayheb (Illinois Institute of Technology) February 15, 2016 Abstract: Distributed storage systems (DSSs) store large amount of data and make it accessible online, anywhere and anytime. To protect against data loss, the data in DSSs
From playlist Nexus Trimester - 2016 - Fundamental Inequalities and Lower Bounds Theme