Topic
Segre embedding
About: Segre embedding is a research topic. Over the lifetime, 84 publications have been published within this topic receiving 1169 citations. The topic is also known as: Segre imbedding & Segre map.
Papers published on a yearly basis
Papers
TL;DR: In this paper, the authors studied syzygies of the Segre embedding of P(V1)×⋯×P(Vn), and proved two finiteness results.
Abstract: We study syzygies of the Segre embedding of P(V1)×⋯×P(Vn), and prove two finiteness results. First, for fixed p but varying n and Vi, there is a finite list of master p-syzygies from which all other p-syzygies can be derived by simple substitutions. Second, we define a power series fp with coefficients in something like the Schur algebra, which contains essentially all the information of p-syzygies of Segre embeddings (for all n and Vi), and show that it is a rational function. The list of master p-syzygies and the numerator and denominator of fp can be computed algorithmically (in theory). The central observation of this paper is that by considering all Segre embeddings at once (i.e., letting n and the Vi vary) certain structure on the space of p-syzygies emerges. We formalize this structure in the concept of a Δ-module. Many of our results on syzygies are specializations of general results on Δ-modules that we establish. Our theory also applies to certain other families of varieties, such as tangent and secant varieties of Segre embeddings.
128 citations
TL;DR: In this paper, the root space decomposition of a Riemannian symmetric space of compact type and its totally geodesic submanifolds (symmetric subspaces) are described.
Abstract: In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces; this is exemplified by the classification of the totally geodesic submanifolds in the complex quadric Q m : = SO ( m + 2 ) / ( SO ( 2 ) × SO ( m ) ) obtained in the second part of the article. The classification shows that the earlier classification of totally geodesic submanifolds of Q m by Chen and Nagano (see [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745–755]) is incomplete. More specifically, two types of totally geodesic submanifolds of Q m are missing from [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745–755]: The first type is constituted by manifolds isometric to C P 1 × R P 1 ; their existence follows from the fact that Q 2 is (via the Segre embedding) holomorphically isometric to C P 1 × C P 1 . The second type consists of 2-spheres of radius 1 2 10 which are neither complex nor totally real in Q m .
109 citations
Book•
1 Jan 1995
TL;DR: In this article, the authors define Tangent Space, Dimension, Smooth and Singular Points, Analytic Uniformization at Smooth Points, examples of topological knottedness at singular points.
Abstract: 1. Affine Varieties.- 1A. Their Definition, Tangent Space, Dimension, Smooth and Singular Points.- 1B. Analytic Uniformization at Smooth Points, Examples of Topological Knottedness at Singular Points.- 1C. Ox,X a UFD when x Smooth Divisor of Zeroes and Poles of Functions.- 2. Projective Varieties.- 2A. Their Definition, Extension of Concepts from Affine to Projective Case.- 2B. Products, Segre Embedding, Correspondences.- 2C. Elimination Theory, Noether's Normalization Lemma, Density of Zariski-Open Sets.- 3. Structure of Correspondences.- 3A. Local Properties-Smooth Maps, Fundamental Openness Principle, Zariski's Main Theorem.- 3B. Global Properties-Zariski's Connectedness Theorem, Specialization Principle.- 3C. Intersections on Smooth Varieties.- 4. Chow's Theorem.- 4A. Internally and Externally Defined Analytic Sets and their Local Descriptions as Branched Coverings of ?n.- 4B. Applications to Uniqueness of Algebraic Structure and Connectedness.- 5. Degree of a Projective Variety.- 5A. Definition of deg X, multxX, of the Blow up Bx(X), Effect of a Projection, Examples.- 5B. Bezout's Theorem.- 5C. Volume of a Projective Variety Review of Homology, DeRham's Theorem, Varieties as Minimal Submanifolds.- 6. Linear Systems.- 6A. The Correspondence between Linear Systems and Rational Maps, Examples Complete Linear Systems are Finite-Dimensional.- 6B. Differential Forms, Canonical Divisors and Branch Loci.- 6C. Hilbert Polynomials, Relations with Degree.- Appendix to Chapter 6. The Weil-Samuel Algebraic Theory of Multiplicity.- 7. Curves and Their Genus.- 7A. Existence and Uniqueness of the Non-Singular Model of Each Function Field of Transcendence Degree 1 (after Albanese).- 7B. Arithmetic Genus = Topological Genus Existence of Good Projections to ?1, ?2, ?3.- 7C. Residues of Differentials on Curves, the Classical Riemann-Roch Theorem for Curves and Applications.- 7D. Curves of Genus 1 as Plane Cubics and as Complex Tori ?/L.- 8. The Birational Geometry of Surfaces.- 8A. Generalities on Blowing up Points.- 8B. Resolution of Singularities of Curves on a Smooth Surface by Blowing up the Surface Examples.- 8C. Factorization of Birational Maps between Smooth Surfaces the Trees of Infinitely Near Points.- 8D. The Birational Map between ?2 and the Quadric and Cubic Surfaces the 27 Lines on a Cubic Surface.- List of Notations.
85 citations
TL;DR: In this paper, it was shown that X ⊂ P n is a reduced and irreducible projective variety, and the equations describing the higher secant varieties of X can be found in the following cases: • X = P n 1 × ⋯ × P n t × p n n, where n is large with respect to the n i (Theorem 2.4).
69 citations
Posted Content•
TL;DR: In this article, it was shown that a product of $m>5$ copies of $PP^1, embedded in the projective space by the standard Segre embedding, is $k$-identifiable (i.e. a general point of the secant variety $S^k(X)$ is contained in only one $(k+1)$-secant $k-space), for all $k $ such that $k+ 1\leq 2m-1}/m
Abstract: We prove that a product of $m>5$ copies of $\PP^1$, embedded in the projective space $\PP^r$ by the standard Segre embedding, is $k$-identifiable (i.e. a general point of the secant variety $S^k(X)$ is contained in only one $(k+1)$-secant $k$-space), for all $k$ such that $k+1\leq 2^{m-1}/m$.
32 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 3 |
| 2020 | 4 |
| 2019 | 2 |
| 2018 | 7 |
| 2017 | 8 |
| 2016 | 4 |