Algebraic varieties

Complete variety

In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism is a closed map (i.e. maps closed sets onto closed sets). This can be seen as an analogue of compactness in algebraic geometry: a topological space X is compact if and only if the above projection map is closed with respect to topological products. The image of a complete variety is closed and is a complete variety. A closed subvariety of a complete variety is complete. A complex variety is complete if and only if it is compact as a complex-analytic variety. The most common example of a complete variety is a projective variety, but there do exist complete non-projective varieties in dimensions 2 and higher. While any complete nonsingular surface is projective, there exist nonsingular complete varieties in dimension 3 and higher which are not projective. The first examples of non-projective complete varieties were given by Masayoshi Nagata and Heisuke Hironaka. An affine space of positive dimension is not complete. The morphism taking a complete variety to a point is a proper morphism, in the sense of scheme theory. An intuitive justification of "complete", in the sense of "no missing points", can be given on the basis of the valuative criterion of properness, which goes back to Claude Chevalley. (Wikipedia).

Video thumbnail

Stereolab - Need To Be

From the album Margerine Eclipse

From playlist the absolute best of stereolab

Video thumbnail

Stereolab - The Super-It

Created with mp32tube.com

From playlist the absolute best of stereolab

Video thumbnail

Stereolab - With Friends Like These

Stereolab - With Friends Like These

From playlist the absolute best of stereolab

Video thumbnail

Schemes 26: Abstract and projective varieties

This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. We discuss the relation between abstract, projective, and complete varieties, and given an example found by Hironaka of a complete variety that is not projecti

From playlist Algebraic geometry II: Schemes

Video thumbnail

William Simmons 4/24/15 Part 1

Title: A Differential Algebra Sampler

From playlist Spring 2015

Video thumbnail

Algebraic and Convex Geometry of Sums of Squares on Varieties (Lecture 3) by Greg Blekherman

PROGRAM COMBINATORIAL ALGEBRAIC GEOMETRY: TROPICAL AND REAL (HYBRID) ORGANIZERS Arvind Ayyer (IISc, India), Madhusudan Manjunath (IITB, India) and Pranav Pandit (ICTS-TIFR, India) DATE: 27 June 2022 to 08 July 2022 VENUE: Madhava Lecture Hall and Online Algebraic geometry is the study of

From playlist Combinatorial Algebraic Geometry: Tropical and Real (HYBRID)

Video thumbnail

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

Video thumbnail

Operational K-theory - Sam Payne

Sam Payne March 13, 2015 Workshop on Chow groups, motives and derived categories More videos on http://video.ias.edu

From playlist Mathematics

Video thumbnail

Arithmetic models for Shimura varieties – Georgios Pappas – ICM2018

Number Theory | Algebraic and Complex Geometry Invited Lecture 3.8 | 4.11 Arithmetic models for Shimura varieties Georgios Pappas Abstract: We describe recent work on the construction of well-behaved arithmetic models for large classes of Shimura varieties and report on progress in the s

From playlist Algebraic & Complex Geometry

Video thumbnail

Ana Caraiani - 3/3 Shimura Varieties and Modularity

We discuss vanishing theorems for the cohomology of Shimura varieties with torsion coefficients, under a genericity condition at an auxiliary prime. We describe two complementary approaches to these results, due to Caraiani-Scholze and Koshikawa, both of which rely on the geometry of the H

From playlist 2022 Summer School on the Langlands program

Video thumbnail

The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties - Will Sawin

Joint IAS/Princeton University Number Theory Seminar Topic: The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties Speaker: Will Sawin Affiliation: Columbia University Date: March 18, 2021 For more video please visit http://video.ias.edu

From playlist Mathematics

Video thumbnail

Whole Number Playlist Intro

This is just a placeholder to introduce all the videos in this playlist.https://www.youtube.com/playlist?list=PLvolZqLMhJmnOJSzXv2I7DDu2aRa-FagD If you have any questions, please contact me at dhabecker@gmail.com

From playlist All About Whole Numbers

Video thumbnail

Low Algebraic Dimension Matrix Completion -Laura Balzano

Virtual Workshop on Missing Data Challenges in Computation Statistics and Applications Topic: Low Algebraic Dimension Matrix Completion Speaker: Laura Balzano Date: September 11, 2020 For more video please visit http://video.ias.edu

From playlist Mathematics

Related pages

Theorem of the cube | Chow's lemma | Compact space | Topological space | Dimension of an algebraic variety | Claude Chevalley | Hironaka's example | Affine space | Mathematics | Proper morphism | Projective variety | Product (category theory) | Algebraic geometry | Fano variety | Algebraic variety | Closed set