On the Terracini Locus of Projective Varieties

Milan Journal of Mathematics ◽

10.1007/s00032-020-00324-5 ◽

2021 ◽

Author(s):

Edoardo Ballico ◽

Luca Chiantini

Keyword(s):

Linear System ◽

Symmetric Product ◽

Secant Varieties ◽

Finite Sets ◽

Projective Varieties ◽

Special Position ◽

Double Points ◽

Veronese Variety ◽

Interpolation Problems

AbstractWe introduce and study properties of the Terracini locus of projective varieties X, which is the locus of finite sets $$S \subset X$$ S ⊂ X such that 2S fails to impose independent conditions to a linear system L. Terracini loci are relevant in the study of interpolation problems over double points in special position, but they also enter naturally in the study of special loci contained in secant varieties to projective varieties.We find some criteria which exclude that a set S belongs to the Terracini locus. Furthermore, in the case where X is a Veronese variety, we bound the dimension of the Terracini locus and we determine examples in which the locus has codimension 1 in the symmetric product of X.

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The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

Mathematics ◽

10.3390/math6120314 ◽

2018 ◽

Vol 6 (12) ◽

pp. 314 ◽

Cited By ~ 8

Author(s):

Alessandra Bernardi ◽

Enrico Carlini ◽

Maria Catalisano ◽

Alessandro Gimigliano ◽

Alessandro Oneto

Keyword(s):

Algebraic Geometry ◽

Projective Variety ◽

Tensor Decomposition ◽

Tensor Rank ◽

Segre Variety ◽

Secant Varieties ◽

Open Problems ◽

Veronese Variety ◽

Strong Motivation

We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

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On the secant varieties to the tangent developable of a Veronese variety

Journal of Algebra ◽

10.1016/j.jalgebra.2005.03.031 ◽

2005 ◽

Vol 288 (2) ◽

pp. 279-286 ◽

Cited By ~ 5

Author(s):

E. Ballico

Keyword(s):

Secant Varieties ◽

Veronese Variety ◽

Tangent Developable

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ON DIFFERENCES BETWEEN THE BORDER RANK AND THE SMOOTHABLE RANK OF A POLYNOMIAL

Glasgow Mathematical Journal ◽

10.1017/s0017089514000378 ◽

2014 ◽

Vol 57 (2) ◽

pp. 401-413 ◽

Cited By ~ 7

Author(s):

WERONIKA BUCZYŃSKA ◽

JAROSŁAW BUCZYŃSKI

Keyword(s):

General Point ◽

Secant Varieties ◽

Secant Variety ◽

Veronese Variety ◽

Border Rank ◽

Veronese Varieties ◽

Special Points

AbstractWe consider higher secant varieties to Veronese varieties. Most points on the rth secant variety are represented by a finite scheme of length r contained in the Veronese variety – in fact, for a general point, the scheme is just a union of r distinct points. A modern way to phrase it is: the smoothable rank is equal to the border rank for most polynomials. This property is very useful for studying secant varieties, especially, whenever the smoothable rank is equal to the border rank for all points of the secant variety in question. In this note, we investigate those special points for which the smoothable rank is not equal to the border rank. In particular, we show an explicit example of a cubic in five variables with border rank 5 and smoothable rank 6. We also prove that all cubics in at most four variables have the smoothable rank equal to the border rank.

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A theorem on Newton's polygon

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100012470 ◽

1934 ◽

Vol 30 (3) ◽

pp. 287-296

Author(s):

R. Frith

Keyword(s):

Linear System ◽

A Priori ◽

Newton Polygon ◽

Base Point ◽

Hyperelliptic Curves ◽

Convex Polygons ◽

Double Points ◽

Base Points ◽

Double Base

Castelnuovo has shown that the maximum freedom of a linear system of curves of genus p is 3p + 5, and that a system with this maximum freedom consists of hyperelliptic curves which can be transformed into a system of curves of order n with an (n − 2)-ple base point and a certain number of double base points; the only exceptions being that when p = 3 the system may be transformable into that of all quartics, and when p = 1 the system may be transformable into that of all cubics. Further, since, in the transformed system, the characteristic series is a it is non-special and hence the redundancy of the base points is zero, therefore each of the double points of this system reduces the freedom by exactly three; and hence if we remove all the double points we get a system of curves of genus p′ and freedom r′ = 3p′ + 5 with only one base point. If we take this point for origin the system of curves can be represented by a single Newton polygon containing in its interior exactly p points, collinear since the curves are hyperelliptic (p ≠ 3), and containing on its boundary 2p + 6 points. From this we can deduce immediately a theorem concerning convex polygons drawn on squared paper; I shall now give an a priori proof of this theorem.

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Divisor Varieties of Symmetric Products

International Mathematics Research Notices ◽

10.1093/imrn/rnaa313 ◽

2021 ◽

Author(s):

John Sheridan

Keyword(s):

Algebraic Curves ◽

Linear Series ◽

Symmetric Product ◽

Fixed Degree ◽

Symmetric Products ◽

Projective Varieties ◽

General Curve ◽

Higher Dimensional ◽

Detailed Picture

Abstract The geometry of divisors on algebraic curves has been studied extensively over the years. The foundational results of this Brill-Noether theory imply that on a general curve, the spaces parametrizing linear series (of fixed degree and dimension) are smooth, irreducible projective varieties of known dimension. For higher dimensional varieties, the story is less well understood. Our purpose in this paper is to study in detail one class of higher dimensional examples where one can hope for a quite detailed picture, namely (the spaces parametrizing) divisors on the symmetric product of a curve.

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Hilbert series of residual intersections

Compositio Mathematica ◽

10.1112/s0010437x15007289 ◽

2015 ◽

Vol 151 (9) ◽

pp. 1663-1687 ◽

Cited By ~ 3

Author(s):

Marc Chardin ◽

David Eisenbud ◽

Bernd Ulrich

Keyword(s):

Hilbert Series ◽

Secant Varieties ◽

Explicit Formulas ◽

Projective Varieties

We give explicit formulas for the Hilbert series of residual intersections of a scheme in terms of the Hilbert series of its conormal modules. In a previous paper, we proved that such formulas should exist. We give applications to the number of equations defining projective varieties and to the dimension of secant varieties of surfaces and three-folds.

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Meaning as a nonlinear effect

AILA Review ◽

10.1075/aila.28.01blo ◽

2015 ◽

Vol 28 ◽

pp. 7-27 ◽

Cited By ~ 11

Author(s):

Jan Blommaert

Keyword(s):

Linear System ◽

Nonlinear Effect ◽

Open Systems ◽

Finite Sets ◽

Units Of Analysis ◽

Linguistic Fact

Saussurean and Chomskyan “conduit” views of meaning in communication, dominant in much of expert and lay linguistic semantics, presuppose a simple, closed and linear system in which outcomes can be predicted and explained in terms of finite sets of rules. Summarizing critical traditions of scholarship, notably those driven by Bateson’s view of systems infused with more recent linguistic-anthropological insights into the ideologically mediated and indexically organized “total linguistic fact”, this paper argues for a view of meaning in terms of complex open systems in which complex units of analysis invite more precise distinctions within “meaning”. Using online viral memes and the metapragmatic qualifier of “cool” as cases in point, we see that the meaning of such memes is better described as a range of “effects”, most of them nonlinear and not predictable on the basis of the features of the sign itself. Such effects suggest a revised and broader notion of nonlinear “perlocution”.

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