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Edoardo Ballico

Publications and source records attributed to Edoardo Ballico.

At least 73 records · Page 4Linked to original sources
arXiv2019

Entry loci and ranks

We study entry loci of varieties and their irreducibility from the perspective of $X$-ranks with respect to a projective variety $X$. These loci are the closures of the points that appear in an $X$-rank decomposition of a general point in the ambient space. We look at entry loci of low degree normal surfaces in $\mathbb P^4$ using Segre points of curves; the smooth case was classically studied by Franchetta. We introduce a class of varieties whose generic rank coincides with the one of its general entry locus, and show that any smooth and irreducible projective variety admits an embedding with this property.

math.AG↗
arXiv2019

Labels of real projective varieties

Let $X$ be a complex projective variety defined over $\mathbb R$. Recently, Bernardi and the first author introduced the notion of admissible rank with respect to $X$. This rank takes into account only decompositions that are stable under complex conjugation. Such a decomposition carries a label, i.e., a pair of integers recording the cardinality of its totally real part. We study basic properties of admissible ranks for varieties, along with special examples of curves; for instance, for rational normal curves admissible and complex ranks coincide. Along the way, we introduce the scheme theoretic version of admissible rank. Finally, analogously to the situation of real ranks, we analyze typical labels, i.e., those arising as labels of a full-dimensional Euclidean open set. We highlight similarities and differences with typical ranks.

math.AG↗
arXiv2019

On the weights of dual codes arising from the GK curve

In this paper we investigate some dual algebraic-geometric codes associated with the Giulietti-Korchmáros maximal curve. We compute the minimum distance and the minimum weight codewords of such codes and we investigate the generalized hamming weights of such codes.

cs.IT↗
arXiv2019

Strict inclusions of high rank loci

For a given projective variety $X$, the high rank loci are the closures of the sets of points whose $X$-rank is higher than the generic one. We show examples of strict inclusion between two consecutive high rank loci. Our first example is for the Veronese surface of plane quartics. Although Piene had already shown an example when $X$ is a curve, we construct infinitely many curves in $\mathbb P^4$ for which such strict inclusion appears. For space curves, we give two criteria to check whether the locus of points of maximal rank 3 is finite (possibly empty).

math.AG↗
arXiv2019

On the Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ of relatively high degree

Let $\mathcal{H}_{d,g,r}$ be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree $d$ and genus $g$ in $\PP^r$. We denote by $\mathcal{H}^\mathcal{L}_{d,g,r}$ the union of those components of $\mathcal{H}_{d,g,r}$ whose general element is linearly normal and we show that any non-empty $\mathcal{H}^\mathcal{L}_{d,g,r}$ ($d\ge g+r-3$) is irreducible for an extensive range of triples $(d,g,r)$ beyond the Brill-Noether range. This establishes the validity of a suitably modified assertion of Severi regarding the irreducibility of the Hilbert scheme $\mathcal{H}^\mathcal{L}_{d,g,r}$ of linearly normal curves for $g+r-3\le d\le g+r$, $r\ge 3$, and $g \ge 2r+3$ if $d=g+r-3$.

math.AG↗
arXiv2019

Minimal degree equations for curves and surfaces (variations on a theme of Halphen)

Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such collections of varieties: their minimal degree and their maximal number of linearly independent smallest degree hypersurfaces passing through them. We show results for curves and surfaces, and pose several questions.

math.AG↗
arXiv2019

Partially complex ranks for real projective varieties

Let $X(\mathbb {C})\subset \mathbb {P}^r(\mathbb {C})$ be an integral non-degenerate variety defined over $\mathbb {R}$. For any $q\in \mathbb {P}^r(\mathbb {R})$ we study the existence of $S\subset X(\mathbb {C})$ with small cardinality, invariant for the complex conjugation and with $q$ contained in the real linear space spanned by $S$. We discuss the advantages of these additive decompositions with respect to the $X(\mathbb {R})$-rank, i.e. the rank of $q$ with respect to $X(\mathbb {R})$. We describe the case of hypersurfaces and Veronese varieties.

math.AG↗
arXiv2019

Dependent subsets of embedded projective varieties

Let $X\subset \mathbb {P}^r$ be an integral and non-degenerate variety. Set $n:= \dim (X)$. Let $ρ(X)''$ be the maximal integer such that every zero-dimensional scheme $Z\subset X$ smoothable in $X$ is linearly independent. We prove that $X$ is linearly normal if $ρ(X)''\ge \lceil (r+2)/2\rceil$ and that $ρ(X)'' < 2\lceil (r+1)/(n+1)\rceil$, unless either $n=r$ or $X$ is a rational normal curve.

math.AG↗
arXiv2019

On the irreducibility of the Severi variety of nodal curves in a smooth surface

Let $X$ be a smooth projective surface and $L\in \mathrm{Pic}(X)$. We prove that if $L$ is $(2k-1)$-spanned, then the set $\tilde{V}_k(L)$ of all nodal and irreducible $D\in |L|$ with exactly $k$ nodes is irreducible. The set $\tilde{V}_k(L)$ is an open subset of a Severi variety of $|L|$, the full Severi variety parametrizing all integral $D\in |L|$ with geometric genus $g(L)-k$.

math.AG↗
arXiv2019

Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails

Let $X\subset \mathbb {P}^r$ be an integral and non-degenerate variety. For any $q\in \mathbb {P}^r$ let $r_X(q)$ be its $X$-rank and $\mathcal {S} (X,q)$ the set of all finite subsets of $X$ such that $|S|=r_X(q)$ and $q\in \langle S\rangle$, where $\langle \ \ \rangle$ denotes the linear span. We consider the case $|\mathcal {S} (X,q)|>1$ (i.e. when $q$ is not $X$-identifiable) and study the set $W(X)_q:= \cap _{S\in\mathcal {S}}\langle S\rangle$, which we call the non-uniqueness set of $q$. We study the case $\dim X=1$ and the case $X$ a Veronese embedding of $\mathbb {P}^n$.

math.AG↗
arXiv2019

On the partially symmetric rank of tensor products of W-states and other symmetric tensors

Given tensors $T$ and $T'$ of order $k$ and $k'$ respectively, the tensor product $T \otimes T'$ is a tensor of order $k+k'$. It was recently shown that the tensor rank can be strictly submultiplicative under this operation ([Christandl-Jensen-Zuiddam]). We study this phenomenon for symmetric tensors where additional techniques from algebraic geometry become available. The tensor product of symmetric tensors results in a partially symmetric tensor and our results amount to bounds on the partially symmetric rank. Following motivations from algebraic complexity theory and quantum information theory, we focus on the so-called "W-states", namely monomials of the form $x^{d-1}y$, and on products of such. In particular, we prove that the partially symmetric rank of $x^{d_1 -1}y \otimes ... \otimes x^{d_k-1} y$ is at most $2^{k-1}(d_1+ ... +d_k)$.

math.AG↗
arXiv2019

Algebraic surfaces with infinitely many twistor lines

We prove that a reduced and irreducible algebraic surface in $\mathbb{CP}^{3}$ containing infinitely many twistor lines cannot have odd degree. Then, exploiting the theory of quaternionic slice regularity and the normalization map of a surface, we give constructive existence results for even degrees.

math.DG↗
arXiv2019

Hypersurface arrangements of aCM type

We investigate the arrangement of hypersurfaces on a nonsingular varieties whose associated logarithmic vector bundle is arithmetically Cohen-Macaulay (for short, aCM), and prove that the projective space is the only smooth complete intersection with Picard rank one that admits an aCM logarithmic vector bundle. We also obtain a number of results on aCM logarithmic vector bundles over several specific varieties. As an opposite situation we investigate the Torelli-type problem that the logarithmic cohomology determines the arrangement.

math.AG↗
arXiv2018

An effective criterion for the additive decompositions of forms

We give an effective criterion for the identifiability of additive decompositions of homogeneous forms of degree $d$ in a fixed number of variables. Asymptotically for large $d$ it has the same order of the Kruskal's criterion adapted to symmetric tensors given by L. Chiantini, G. Ottaviani and N. Vannieuwenhoven. We give a new case of indentifiability for $d=4$.

math.AG↗
arXiv2018

Twistor lines on algebraic surfaces

We give quantitative and qualitative results on the family of surfaces in $\mathbb{CP}^3$ containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines $E$. We prove that its general element is a smooth surface containing $E$ and no other line. Afterwards we prove that twistor lines are Zariski dense in the Grassmannian $Gr(2,4)$. Then, for any degree $d\ge 4$, we give lower bounds on the maximum number of twistor lines contained in a degree $d$ surface. The smooth and singular cases are studied as well as the $j$-invariant one.

math.AG↗
arXiv2018

Three Topological Results on the Twistor Discriminant Locus in the 4-Sphere

We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration $π:\mathbb{CP}^{3}\to S^{4}$. We prove three results about the topology of the twistor discriminant locus of an algebraic surface in $\mathbb{CP}^{3}$. First of all we prove that, with the exception of two exceptional cases, the real dimension of the twistor discriminant locus of an algebraic surface is always equal to 2. Secondly we describe the possible intersections of a general surface with the family of twistor lines: we find that only 4 configurations are possible and for each of them we compute the dimension. Lastly we give a decomposition of the twistor discriminant locus of a given cone in terms of its singular locus and its dual variety.

math.DG↗
arXiv2018

Representation type of surfaces in $\mathbb{P}^3$

The goal of this article is to prove that every surface with a regular point in the three-dimensional projective space of degree at least four, is of wild representation type under the condition that either $X$ is integral or $\mathrm{Pic}(X) \cong \langle \Oo_X(1) \rangle$; we construct families of arbitrarily large dimension of indecomposable pairwise non-isomorphic aCM vector bundles. On the other hand, we prove that every non-integral aCM scheme of arbitrary dimension at least two, is also very wild in a sense that there exist arbitrarily large dimensional families of pairwise non-isomorphic aCM non-locally free sheaves of rank one.

math.AG↗