Vector bundles | Algebraic geometry | Geometry of divisors
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety X amounts to understanding the different ways of mapping X into projective space. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor. In more detail, a line bundle is called basepoint-free if it has enough sections to give a morphism to projective space. A line bundle is semi-ample if some positive power of it is basepoint-free; semi-ampleness is a kind of "nonnegativity". More strongly, a line bundle on X is very ample if it has enough sections to give a closed immersion (or "embedding") of X into projective space. A line bundle is ample if some positive power is very ample. An ample line bundle on a projective variety X has positive degree on every curve in X. The converse is not quite true, but there are corrected versions of the converse, the Nakai–Moishezon and Kleiman criteria for ampleness. (Wikipedia).
Ampleness in strongly minimal structures - K. Tent - Workshop 3 - CEB T1 2018
Katrin Tent (Münster) / 30.03.2018 Ampleness in strongly minimal structures The notion of ampleness captures essential properties of projective spaces over fields. It is natural to ask whether any sufficiently ample strongly minimal set arises from an algebraically closed field. In this
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
Free ebook http://tinyurl.com/EngMathYT How to integrate over 2 curves. This example discusses the additivity property of line integrals (sometimes called path integrals).
From playlist Engineering Mathematics
Introduction to Fiber Bundles part 1: Definitions
We give the definition of a fiber bundle with fiber F, trivializations and transition maps. This is a really basic stuff that we use a lot. Here are the topics this sets up: *Associated Bundles/Principal Bundles *Reductions of Structure Groups *Steenrod's Theorem *Torsor structure on arith
From playlist Fiber bundles
Calculus 3: Line Integrals (4 of 44) What is a Line Integral? NOT TO BE CONFUSED WITH
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the line integral is NOT the length of a line (curve). Next video in the series can be seen at: https://youtu.be/yUHGDBYxGe0
From playlist CALCULUS 3 CH 6 LINE INTEGRALS
Physics Ch 67.1 Advanced E&M: Review Vectors (48 of TBD) What is a Line Integral?
Visit http://ilectureonline.com for more math and science lectures! To donate: http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will learn that a line integral is the integral of the dot product of a vector and an infinitesimal displacement along a path from
From playlist PHYSICS 67.1 ADVANCED E&M VECTORS & FIELDS
Free ebook http://tinyurl.com/EngMathYT Solution to a line integral problem featuring integration over curves. Such ideas have important applications in applied mathematics, physics and engineering.
From playlist Engineering Mathematics
Introduction to Line Integrals
Introduction to Line Integrals If you enjoyed this video please consider liking, sharing, and subscribing. You can also help support my channel by becoming a member https://www.youtube.com/channel/UCr7lmzIk63PZnBw3bezl-Mg/join Thank you:)
From playlist Calculus 3
Improper Integral with Two Infinite Limits
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Improper Integral with Two Infinite Limits
From playlist Calculus
A. Höring - A decomposition theorem for singular spaces with trivial canonical class (Part 3)
The Beauville-Bogomolov decomposition theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, an irreducible, simply-connected Calabi-Yau, and holomorphic symplectic manifolds. With the deve
From playlist Ecole d'été 2019 - Foliations and algebraic geometry
Juliette Bruce - Semi-Ample Asymptotic Syzygies - WAGON
I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-am
From playlist WAGON
Projectivity of the moduli space of KSBA stable pairs and applications - Zsolt Patakfalvi
Zsolt Patakfalvi Princeton University February 24, 2015 KSBA (Kollár-Shepherd-Barron-Alexeev) stable pairs are higher dimensional generalizations of (weighted) stable pointed curves. I will present a joint work in progress with Sándor Kovács on proving the projectivity of this moduli spac
From playlist Mathematics
D. Brotbek - On the hyperbolicity of general hypersurfaces
A smooth projective variety over the complex numbers is said to be (Brody) hyperbolic if it doesn’t contain any entire curve. Kobayashi conjectured in the 70’s that general hypersurfaces of sufficiently large degree in PN are hyperbolic. This conjecture was only recently proved by Siu. Th
From playlist Complex analytic and differential geometry - a conference in honor of Jean-Pierre Demailly - 6-9 juin 2017
Cécile Gachet : Positivity of higher exterior powers of the tangent bundle
CONFERENCE Recording during the thematic meeting : "Algebraic Geometry and Complex Geometry " the December1, 2022 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Jean Petit Find this video and other talks given by worldwide mathematicians on CIRM's
From playlist Algebraic and Complex Geometry
Hypotheses in Geometric Versions of Diophantine Problems
Here describe the notion of isotriviality and how it plays roles in the geometric versions of Mordell-Lang and Lang-Bombieri-Noguchi.
From playlist Mordell-Lang
Schemes 42: Very ample sheaves
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. We define ample and very ample invertible sheaves for projective varieties, and gives some examples for complex elliptic curves. We also show that some sect
From playlist Algebraic geometry II: Schemes
Low degree points on curves. - Vogt - Workshop 2 - CEB T2 2019
Isabel Vogt (MIT) / 27.06.2019 Low degree points on curves. In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris–S
From playlist 2019 - T2 - Reinventing rational points
R. Lazarsfeld: The Equations Defining Projective Varieties. Part 3.2
The lecture was held within the framework of the Junior Hausdorff Trimester Program Algebraic Geometry. (14.1.2014)
From playlist HIM Lectures: Junior Trimester Program "Algebraic Geometry"
15_1 An Introduction to Line Integrals
In this lecture I start to explain the concept of a line integral.
From playlist Advanced Calculus / Multivariable Calculus