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26 pages, 434 KB

Open AccessFeature PaperArticle

On the Hilbert Function of Partially General Unions of Double Points

by Edoardo Ballico

Mathematics 2025, 13(23), 3808; https://doi.org/10.3390/math13233808 - 27 Nov 2025

Viewed by 178

We study the Hilbert function of a union of a fixed set of double points and a prescribed number of general double points. Our main results are for double points of Veronese varieties and of Segre–Veronese varieties. Among the Segre–Veronese varieties, we study [...] Read more.

We study the Hilbert function of a union of a fixed set of double points and a prescribed number of general double points. Our main results are for double points of Veronese varieties and of Segre–Veronese varieties. Among the Segre–Veronese varieties, we study the ones with all factors of dimension one and the ones with two factors, one of them of dimension one. We give many examples with exceptional or controlled behavior for a small number of double points. Full article

17 pages, 333 KB

Open AccessArticle

The Next Terracini Loci of Segre–Veronese Varieties and Their Maximal Weights

by Edoardo Ballico

Mathematics 2025, 13(19), 3166; https://doi.org/10.3390/math13193166 - 2 Oct 2025

Viewed by 529

Abstract

We describe all Terracini loci of Segre–Veronese varieties with at most roughly double the points of the minimal one. In this range we compute the maximum of all weights of the Terracini sets. To prove these results we use cohomological tools (residual exact [...] Read more.

We describe all Terracini loci of Segre–Veronese varieties with at most roughly double the points of the minimal one. In this range we compute the maximum of all weights of the Terracini sets. To prove these results we use cohomological tools (residual exact sequences) applied to some critical schemes associated with a Terracini set and containing all of its points. We expect that these critical schemes will be a very useful tool for other related problems. Full article

16 pages, 343 KB

Open AccessArticle

Tame Secant Varieties and Group Actions

by Edoardo Ballico

Axioms 2025, 14(7), 542; https://doi.org/10.3390/axioms14070542 - 20 Jul 2025

Viewed by 460

Abstract

Let X be a complex projective variety embedded in a complex projective space. The dimensions of the secant varieties of X have an expected value, and it is important to know if they are equal or at least near to this expected value. [...] Read more.

Let X be a complex projective variety embedded in a complex projective space. The dimensions of the secant varieties of X have an expected value, and it is important to know if they are equal or at least near to this expected value. Blomenhofer and Casarotti proved important results on the embeddings of G-varieties, G being an algebraic group, embedded in the projectivations of an irreducible G-representation, proving that no proper secant variety is a cone. In this paper, we give other conditions which assure that no proper secant varieties of X are a cone, e.g., that X is G-homogeneous. We consider the Segre product of two varieties with the product action and the case of toric varieties. We present conceptual tests for it, and discuss the information we obtained from certain linear projections of X. For the Segre–Veronese embeddings of

P^{n} \times P^{n}

with respect to forms of bidegree

(1, d)

, our results are related to the simultaneous rank of degree d forms in

n + 1

variables. Full article

21 pages, 395 KB

Open AccessArticle

Interpolation of Polynomials and Singular Curves: Segre and Veronese Varieties

by Edoardo Ballico

Symmetry 2024, 16(12), 1683; https://doi.org/10.3390/sym16121683 - 19 Dec 2024

Viewed by 1045

Abstract

We study an interpolation problem (objects singular at a prescribed finite set) for curves instead of hypersurfaces. We study singular curves in projective and multiprojective spaces. We construct curves that are singular (or with maximal dimension Zariski tangent space) at each point of [...] Read more.

We study an interpolation problem (objects singular at a prescribed finite set) for curves instead of hypersurfaces. We study singular curves in projective and multiprojective spaces. We construct curves that are singular (or with maximal dimension Zariski tangent space) at each point of a prescribed finite set, while the curves have low degree or low “complexity” (e.g., they are complete intersections of hypersurfaces of low degree). We discuss six open problems on the existence and structure of the base locus of the set of all hypersurfaces of a given degree and singular at a prescribed number of general points. The tools come from algebraic geometry, and some of the results are only existence ones or only asymptotic ones (but with as explicit as possible bounds). Some of the existence results are almost constructive, i.e., in our framework, random parameters should give a solution, or otherwise, take other random parameters. Full article

(This article belongs to the Section Mathematics)

14 pages, 337 KB

Open AccessFeature PaperArticle

Terracini Loci: Dimension and Description of Its Components

by Edoardo Ballico

Mathematics 2023, 11(22), 4702; https://doi.org/10.3390/math11224702 - 20 Nov 2023

Cited by 2 | Viewed by 1249

Abstract

We study the Terracini loci of an irreducible variety X embedded in a projective space: non-emptiness, dimensions and the geometry of their maximal dimension’s irreducible components. These loci were studied because they describe where the differential of an important geometric map drops rank. [...] Read more.

We study the Terracini loci of an irreducible variety X embedded in a projective space: non-emptiness, dimensions and the geometry of their maximal dimension’s irreducible components. These loci were studied because they describe where the differential of an important geometric map drops rank. Our best results are if X is either a Veronese embedding of a projective space of arbitrary dimension (the set-up for the additive decomposition of homogeneous polynomials) or a Segre–Veronese embedding of a multiprojective space (the set-up for partially symmetric tensors). For an arbitrary X, we give several examples in which all Terracini loci are empty, several criteria for non-emptiness and examples with the maximal defect possible a priori of an element of a minimal Terracini locus. We raise a few open questions. Full article

16 pages, 361 KB

Open AccessArticle

Base Point Freeness, Uniqueness of Decompositions and Double Points for Veronese and Segre Varieties

by Edoardo Ballico

Symmetry 2021, 13(12), 2344; https://doi.org/10.3390/sym13122344 - 6 Dec 2021

Viewed by 2569

Abstract

We prove a base point freeness result for linear systems of forms vanishing at general double points of the projective plane. For tensors we study the uniqueness problem for the representation of a tensor as a sum of terms corresponding to points and [...] Read more.

We prove a base point freeness result for linear systems of forms vanishing at general double points of the projective plane. For tensors we study the uniqueness problem for the representation of a tensor as a sum of terms corresponding to points and tangent vectors of the Segre variety associated with the format of the tensor. We give complete results for unions of one point and one tangent vector. Full article

(This article belongs to the Special Issue Advances in Symmetric Tensor Decomposition Methods)

86 pages, 894 KB

Open AccessArticle

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

by Alessandra Bernardi, Enrico Carlini, Maria Virginia Catalisano, Alessandro Gimigliano and Alessandro Oneto

Mathematics 2018, 6(12), 314; https://doi.org/10.3390/math6120314 - 8 Dec 2018

Cited by 47 | Viewed by 9222

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Decomposability of Tensors)

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13 pages, 320 KB

Open AccessArticle

On Comon’s and Strassen’s Conjectures

by Alex Casarotti, Alex Massarenti and Massimiliano Mella

Mathematics 2018, 6(11), 217; https://doi.org/10.3390/math6110217 - 25 Oct 2018

Cited by 6 | Viewed by 3356

Abstract

Comon’s conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We [...] Read more.

Comon’s conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon’s conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties. Full article

(This article belongs to the Special Issue Decomposability of Tensors)

9 pages, 265 KB

Open AccessArticle

Set Evincing the Ranks with Respect to an Embedded Variety (Symmetric Tensor Rank and Tensor Rank

by Edoardo Ballico

Mathematics 2018, 6(8), 140; https://doi.org/10.3390/math6080140 - 14 Aug 2018

Viewed by 3929

Abstract

Let

X \subset P^{r}

be an integral and non-degenerate variety. We study when a finite set

S \subset X

evinces the X-rank of the general point of the linear span of S. We give a criterion when X is the [...] Read more.

Let

X \subset P^{r}

be an integral and non-degenerate variety. We study when a finite set

S \subset X

evinces the X-rank of the general point of the linear span of S. We give a criterion when X is the order d Veronese embedding

X_{n, d}

of

P^{n}

and

| S | \leq (\binom{n + ⌊ d / 2 ⌋}{n})

. For the tensor rank, we describe the cases with

| S | \leq 3

. For

X_{n, d},

we raise some questions of the maximum rank for

d ≫ 0

(for a fixed n) and for

n ≫ 0

(for a fixed d). Full article

(This article belongs to the Special Issue Decomposability of Tensors)

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