Math 6130 Notes. Fall 2002. 5. Basic Examples. Before We Go Further With
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Hyperplane Sections of Determinantal Varieties Over Finite Fields And
Hyperplane Sections of Determinantal Varieties over Finite Fields and Linear Codes Peter Beelen∗ Sudhir R. Ghorpade† Abstract We determine the number of Fq-rational points of hyperplane sections of classical deter- minantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify sections giving the maximum number of Fq-rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is closely related to the study of linear codes associated to determi- nantal varieties, and the determination of their weight distribution, minimum distance and generalized Hamming weights. The previously known results about these are generalized and expanded significantly. 1 Introduction The classical determinantal variety defined by the vanishing all minors of a fixed size in a generic matrix is an object of considerable importance and ubiquity in algebra, combinatorics, algebraic geometry, invariant theory and representation theory. The defining equations clearly have integer coefficients and as such the variety can be defined over any finite field. The number of Fq-rational points of this variety is classically known. We are mainly interested in a more challenging question of determining the number of Fq-rational points of such a variety when intersected with a hyperplane in the ambient projective space, or more generally, with a linear subvariety of a fixed codimension in the ambient projective space. In particular, we wish to know which of these sections have the maximum number of Fq-rational points. These questions are directly related to determining the complete weight distribution and the generalized Hamming weights of the associated linear codes, which are caledl determinantal codes. -
Phylogenetic Algebraic Geometry
PHYLOGENETIC ALGEBRAIC GEOMETRY NICHOLAS ERIKSSON, KRISTIAN RANESTAD, BERND STURMFELS, AND SETH SULLIVANT Abstract. Phylogenetic algebraic geometry is concerned with certain complex projec- tive algebraic varieties derived from finite trees. Real positive points on these varieties represent probabilistic models of evolution. For small trees, we recover classical geometric objects, such as toric and determinantal varieties and their secant varieties, but larger trees lead to new and largely unexplored territory. This paper gives a self-contained introduction to this subject and offers numerous open problems for algebraic geometers. 1. Introduction Our title is meant as a reference to the existing branch of mathematical biology which is known as phylogenetic combinatorics. By “phylogenetic algebraic geometry” we mean the study of algebraic varieties which represent statistical models of evolution. For general background reading on phylogenetics we recommend the books by Felsenstein [11] and Semple-Steel [21]. They provide an excellent introduction to evolutionary trees, from the perspectives of biology, computer science, statistics and mathematics. They also offer numerous references to relevant papers, in addition to the more recent ones listed below. Phylogenetic algebraic geometry furnishes a rich source of interesting varieties, includ- ing familiar ones such as toric varieties, secant varieties and determinantal varieties. But these are very special cases, and one quickly encounters a cornucopia of new varieties. The objective of this paper is to give an introduction to this subject area, aimed at students and researchers in algebraic geometry, and to suggest some concrete research problems. The basic object in a phylogenetic model is a tree T which is rooted and has n labeled leaves. -
Segre Embedding and Related Maps and Immersions in Differential Geometry
SEGRE EMBEDDING AND RELATED MAPS AND IMMERSIONS IN DIFFERENTIAL GEOMETRY BANG-YEN CHEN Abstract. Segre embedding was introduced by C. Segre (1863–1924) in his famous 1891 article [50]. The Segre embedding plays an important roles in algebraic geometry as well as in differential geometry, mathemat- ical physics, and coding theory. In this article, we survey main results on Segre embedding in differential geometry. Moreover, we also present recent differential geometric results on maps and immersions which are constructed in ways similar to Segre embedding. Table of Contents.1 1. Introduction. 2. Basic formulas and definitions. 3. Differential geometric characterizations of Segre embedding. 4. Degree of K¨ahlerian immersions and homogeneous K¨ahlerian subman- ifolds via Segre embedding. 5. CR-products and Segre embedding. 6. CR-warped products and partial Segre CR-immersions. 7. Real hypersurfaces as partial Segre embeddings. 8. Complex extensors, Lagrangian submanifolds and Segre embedding. 9. Partial Segre CR-immersions in complex projective space. 10. Convolution of Riemannian manifolds. Arab. J. Math. Sci. 8 (2002), 1–39. 2000 Mathematics Subject Classification. Primary 53-02, 53C42, 53D12; Secondary arXiv:1307.0459v1 [math.AG] 1 Jul 2013 53B25, 53C40, 14E25. Key words and phrases. Segre embedding, totally real submanifold, Lagrangian subman- ifold, CR-submanifold, warped product, CR-warped product, complex projective space, geo- metric inequality, convolution, Euclidean Segre map, partial Segre map, tensor product im- mersion, skew Segre embedding. 1 The author would like to express his thanks to the editors for the invitation to publish this article in this Journal 1 2 BANG-YEN CHEN 11. Convolutions and Euclidean Segre maps. -
Tensor Categorical Foundations of Algebraic Geometry
Tensor categorical foundations of algebraic geometry Martin Brandenburg { 2014 { Abstract Tannaka duality and its extensions by Lurie, Sch¨appi et al. reveal that many schemes as well as algebraic stacks may be identified with their tensor categories of quasi-coherent sheaves. In this thesis we study constructions of cocomplete tensor categories (resp. cocontinuous tensor functors) which usually correspond to constructions of schemes (resp. their morphisms) in the case of quasi-coherent sheaves. This means to globalize the usual local-global algebraic geometry. For this we first have to develop basic commutative algebra in an arbitrary cocom- plete tensor category. We then discuss tensor categorical globalizations of affine morphisms, projective morphisms, immersions, classical projective embeddings (Segre, Pl¨ucker, Veronese), blow-ups, fiber products, classifying stacks and finally tangent bundles. It turns out that the universal properties of several moduli spaces or stacks translate to the corresponding tensor categories. arXiv:1410.1716v1 [math.AG] 7 Oct 2014 This is a slightly expanded version of the author's PhD thesis. Contents 1 Introduction 1 1.1 Background . .1 1.2 Results . .3 1.3 Acknowledgements . 13 2 Preliminaries 14 2.1 Category theory . 14 2.2 Algebraic geometry . 17 2.3 Local Presentability . 21 2.4 Density and Adams stacks . 22 2.5 Extension result . 27 3 Introduction to cocomplete tensor categories 36 3.1 Definitions and examples . 36 3.2 Categorification . 43 3.3 Element notation . 46 3.4 Adjunction between stacks and cocomplete tensor categories . 49 4 Commutative algebra in a cocomplete tensor category 53 4.1 Algebras and modules . 53 4.2 Ideals and affine schemes . -
Characterization of Quantum Entanglement Via a Hypercube of Segre Embeddings
Characterization of quantum entanglement via a hypercube of Segre embeddings Joana Cirici∗ Departament de Matemàtiques i Informàtica Universitat de Barcelona Gran Via 585, 08007 Barcelona Jordi Salvadó† and Josep Taron‡ Departament de Fisíca Quàntica i Astrofísica and Institut de Ciències del Cosmos Universitat de Barcelona Martí i Franquès 1, 08028 Barcelona A particularly simple description of separability of quantum states arises naturally in the setting of complex algebraic geometry, via the Segre embedding. This is a map describing how to take products of projective Hilbert spaces. In this paper, we show that for pure states of n particles, the corresponding Segre embedding may be described by means of a directed hypercube of dimension (n − 1), where all edges are bipartite-type Segre maps. Moreover, we describe the image of the original Segre map via the intersections of images of the (n − 1) edges whose target is the last vertex of the hypercube. This purely algebraic result is then transferred to physics. For each of the last edges of the Segre hypercube, we introduce an observable which measures geometric separability and is related to the trace of the squared reduced density matrix. As a consequence, the hypercube approach gives a novel viewpoint on measuring entanglement, naturally relating bipartitions with q-partitions for any q ≥ 1. We test our observables against well-known states, showing that these provide well-behaved and fine measures of entanglement. arXiv:2008.09583v1 [quant-ph] 21 Aug 2020 ∗ [email protected] † [email protected] ‡ [email protected] 2 I. INTRODUCTION Quantum entanglement is at the heart of quantum physics, with crucial roles in quantum information theory, superdense coding and quantum teleportation among others. -
Equations for Point Configurations to Lie on a Rational Normal Curve
EQUATIONS FOR POINT CONFIGURATIONS TO LIE ON A RATIONAL NORMAL CURVE ALESSIO CAMINATA, NOAH GIANSIRACUSA, HAN-BOM MOON, AND LUCA SCHAFFLER Abstract. The parameter space of n ordered points in projective d-space that lie on a ra- tional normal curve admits a natural compactification by taking the Zariski closure in (Pd)n. The resulting variety was used to study the birational geometry of the moduli space M0;n of n-tuples of points in P1. In this paper we turn to a more classical question, first asked independently by both Speyer and Sturmfels: what are the defining equations? For conics, namely d = 2, we find scheme-theoretic equations revealing a determinantal structure and use this to prove some geometric properties; moreover, determining which subsets of these equations suffice set-theoretically is equivalent to a well-studied combinatorial problem. For twisted cubics, d = 3, we use the Gale transform to produce equations defining the union of two irreducible components, the compactified configuration space we want and the locus of degenerate point configurations, and we explain the challenges involved in eliminating this extra component. For d ≥ 4 we conjecture a similar situation and prove partial results in this direction. 1. Introduction Configuration spaces are central to many modern areas of geometry, topology, and physics. Rational normal curves are among the most classical objects in algebraic geometry. In this paper we explore a setting where these two realms meet: the configuration space of n ordered points in Pd that lie on a rational normal curve. This is naturally a subvariety of (Pd)n, and by taking the Zariski closure we obtain a compactification of this configuration space, which d n we call the Veronese compactification Vd;n ⊆ (P ) . -
Fano 3-Folds in Format, Tom and Jerry
European Journal of Mathematics https://doi.org/10.1007/s40879-017-0200-2 RESEARCH ARTICLE Fano 3-folds in P2×P2 format, Tom and Jerry Gavin Brown1 · Alexander M. Kasprzyk2 · Muhammad Imran Qureshi3 Received: 30 May 2017 / Accepted: 23 October 2017 © The Author(s) 2017. This article is an open access publication Abstract We study Q-factorial terminal Fano 3-folds whose equations are modelled on those of the Segre embedding of P2 ×P2. These lie in codimension 4 in their total anticanonical embedding and have Picard rank 2. They fit into the current state of classification in three different ways. Some families arise as unprojections of degen- erations of complete intersections, where the generic unprojection is a known prime Fano 3-fold in codimension 3; these are new, and an analysis of their Gorenstein projections reveals yet other new families. Others represent the “second Tom” unpro- jection families already known in codimension 4, and we show that every such family contains one of our models. Yet others have no easy Gorenstein projection analysis at all, so prove the existence of Fano components on their Hilbert scheme. Keywords Fano 3-fold · Segre embedding · Gorenstein format Mathematics Subject Classification 14J30 · 14J10 · 14J45 · 14M07 AMK was supported by EPRSC Fellowship EP/N022513/1. MIQ was supported by a Lahore University of Management Sciences (LUMS) faculty startup research grant (STG-MTH-1305) and an LMS research in pairs grant for a visit to the UK. B Gavin Brown [email protected] Alexander M. Kasprzyk [email protected] Muhammad Imran Qureshi [email protected] 1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK 2 School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK 3 Department of Mathematics, SBASSE, LUMS, Lahore 54792, Pakistan 123 G. -
Segre Embedding and Related Maps and Immersions In
Arab J. Math. Sc. Volume 8, Number 1, June 2002 pp. 1-39 SEGRE EMBEDDING AND RELATED MAPS AND IMMERSIONS IN DIFFERENTIAL GEOMETRY BANG-YEN CHEN ABSTRACT. The Segre embedding was introduced by C. Segre (1863-1924) in his famous 1891 article [46]. It plays an important role in algebraic geom etry as well as in differential geometry, mathematical physics, and coding theory. In this article, we survey main results on Segre embeddings in dif ferential geometry. Moreover, we also present recent differential geometric results on maps and immersions which are constructed in ways similar to Segre embedding. TABLE OF CONTENTS1 1. Introduction. 2. Basic formulas and definitions. 3. Differential geometric characterizations of Segre embedding. 4. Degree of Kahlerian immersions and homogeneous Kahlerian subman- ifolds via Segre embedding. 5. OR-products and Segre embedding. 6. OR-warped products and partial Segre OR-immersions. 7. Real hypersurfaces as partial Segre embeddings. 8. Complex extensors, Lagrangian submanifolds and Segre embedding. 9. Partial Segre OR-immersions in complex projective space. 2000 Mathematics Subject Classification. Primary 53-02, 53C42, 53D12; Secondary 53B25, 53C40, 14E25. Key word and phmses. Segre embedding, totally real submanifold, Lagrangian submani fold, CR-submanifold, warped product, CR-warped product, complex projective space, geo metric inequality, convolution, Euclidean Segre map, partial Segre map, tensor product immersion, skew Segre embedding. 1The author would like to express his thanks to the editors for the invitation to publish this article in this Journal 2 BANG-YEN CHEN 10. Convolution of Riemannian manifolds. 11. Convolutions and Euclidean Segre maps. 12. Skew Segre embedding. 13. Tensor product immersions and Segre embedding. -
TRACE TEST Introduction Numerical Algebraic Geometry [11]
Mss., 26 May 2017, arXiv.org/1608.00540 TRACE TEST ANTON LEYKIN, JOSE ISRAEL RODRIGUEZ, AND FRANK SOTTILE Abstract. The trace test in numerical algebraic geometry verifies the completeness of a witness set of an irreducible variety in affine or projective space. We give a brief derivation of the trace test and then consider it for subvarieties of products of projective spaces using multihomogeneous witness sets. We show how a dimension reduction leads to a practical trace test in this case involving a curve in a low-dimensional affine space. Introduction Numerical algebraic geometry [11] uses numerical analysis to study algebraic varieties, which are sets defined by polynomial equations. It is becoming a core tool in applications of algebraic geometry outside of mathematics. Its fundamental concept is a witness set, which is a general linear section of an algebraic variety [8]. This gives a representation of a variety which may be manipulated on a computer and forms the basis for many algorithms. The trace test is used to verify that a witness set is complete. We illustrate this with the folium of Descartes, defined by x3 +y3 = 3xy. A general line ℓ meets the folium in three points W and the pair (W, ℓ) forms a witness set for the folium. Tracking the points of W as ℓ moves computes witness sets on other lines. Figure 1 shows these witness sets on four parallel lines. It also shows the average of each witness set, x3 + y3 = 3xy ℓ ✛ collinear traces Figure 1. Witness sets and the trace test for the folium of Descartes which is one-third of their sum, the trace. -
Arcs on Determinantal Varieties
Arcs on Determinantal Varieties Roi Docampo† June 2, 2018 Abstract We study arc spaces and jet schemes of generic determinantal varieties. Using the natural group action, we decompose the arc spaces into orbits, and analyze their structure. This allows us to compute the number of irreducible components of jet schemes, log canonical thresholds, and topological zeta functions. Introduction Let M = Ars denote the space of r × s matrices, and assume that r ≤ s. Let Dk ⊂ M be the generic deter- minantal variety of rank k, that is, the subvariety of M whose points correspond to matrices of rank at most k. The purpose of this paper is to analyze the structure of arc spaces and jet schemes of generic determinantal varieties. Arc spaces and jet schemes have attracted considerable attention in recent years. They were intro- duced to the field by J. F. Nash [Nas95], who noticed for the fist time their connection with resolution of singularities. A few years later, M. Kontsevich introduced motivic integration [Kon95, DL99], popu- larizing the use of the arc space. And starting with the work of M. Musta¸t˘a, arc and jets have become a standard tool in birational geometry, mainly because of their role in formulas for controlling discrepan- cies [Mus01, Mus02, EMY03, ELM04, EM06, dFEI08]. But despite their significance from a theoretical point of view, arc spaces are often hard to compute in concrete examples. The interest in Nash’s conjecture led to the study of arcs in isolated surface singulari- ties [LJ90, Nob91, LJR98, Plé05, PPP06, LJR08]. Quotient singularities are analyzed from the point of view of motivic integration in [DL02]. -
Evaluation Codes from Smooth Quadric Surfaces and Twisted Segre Varieties Alain Couvreur, Iwan Duursma
Evaluation codes from smooth quadric surfaces and twisted Segre varieties Alain Couvreur, Iwan Duursma To cite this version: Alain Couvreur, Iwan Duursma. Evaluation codes from smooth quadric surfaces and twisted Segre varieties. Designs, Codes and Cryptography, Springer Verlag, 2013, 66 (1-3), pp.291-303. 10.1007/s10623-012-9692-4. hal-00707540 HAL Id: hal-00707540 https://hal.inria.fr/hal-00707540 Submitted on 12 Jun 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Evaluation codes from smooth quadric surfaces and twisted Segre varieties Alain Couvreur∗and Iwan Duursma† May 15, 2012 Abstract We give the parameters of any evaluation code on a smooth quadric surface. For hyperbolic quadrics the approach uses elementary results on product codes and the parameters of codes on elliptic quadrics are obtained by detecting a BCH structure on these codes and using the BCH bound. The elliptic quadric is a twist of the surface P1 × P1 and we detect a similar BCH structure on twists of the Segre embedding of a product of any d copies of the projective line. Keywords: Evaluation codes, Algebraic Geometry codes, quadric surfaces, BCH codes, Segre embed- ding. -
Deformation of Determinantal Schemes Compositio Mathematica, Tome 30, No 3 (1975), P
COMPOSITIO MATHEMATICA DAN LAKSOV Deformation of determinantal schemes Compositio Mathematica, tome 30, no 3 (1975), p. 273-292 <http://www.numdam.org/item?id=CM_1975__30_3_273_0> © Foundation Compositio Mathematica, 1975, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ COMPOSITIO MATHEMATICA, Vol. 30, Fasc. 3, 1975, pag. 273-292 Noordhoff International Publishing Printed in the Netherlands DEFORMATION OF DETERMINANTAL SCHEMES Dan Laksov 1. Introduction We shall in the following work deal with the problem of constructing (global, flat) deformations of determinantal subschemes of affine spaces. Our main contribution is the construction of deformations, such that the generic member of each flat family has a stratification consisting of determinantal schemes, each member of the stratification being the singular locus of the preceding. With the exception of the determinantal schemes of codimension one, a stratification of the above type is generally the best structure that can be obtained. Indeed, T. Svanes has proved (in [17]) that with the above exception, the natural stratification of a generic determinantal variety, given by determinantal subvarieties corresponding to matrices of successively lower rank, is rigid. Hence all members of a family deforming a generic determinantal variety, have the same stratification. The determinantal subschemes of codimension one in an affine space are simpler.