arXiv subjects

Giorgio Ottaviani

Publications and source records attributed to Giorgio Ottaviani.

At least 19 recordsLinked to original sources

arXiv2026

When is one pinhole camera image equal to some other pinhole camera image?

Generically, one expects the images of two different point sets, in two different (projective) cameras, to be different. However, it can happen that the images are the same up to a projective transformation which is an instance of ill-posedness in computer vision. We prove that the images can become projectively equivalent only for point pairs with at most seven elements. In each case, we give explicit descriptions of the Zariski closure of the locus of camera centers which we call the centers-variety. To do this we use classical invariant theory and the geometry of moduli spaces of ordered points in the projective plane. The most involved case is that of seven points which uses a natural parametrization of the Goepel variety.

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arXiv2026

Some Classical Invariants, from Harmonic Quadruples to Triangle Groups

These notes are an expanded version of the lectures held in Tromso, in May 2025 at the "Lie-Stormer Summer School : Invariant Theory from classics to modern developments", in the framework of TiME events. We emphasize the analogy between binary quartics and ternary cubics (and subsequently modular forms) based on their harmonic and equianharmonic invariants. Triangle groups are presented in both the elliptic and the hyperbolic setting with their associated tilings. The topics include the discussion of a short Hilbert paper on polynomials which are powers, that was proposed to the participants. The appendix contains some exercises, with sketches of solutions, and a section devoted to Pfaffians edited by Vincenzo Galgano.

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arXiv2025

Symmetric Persistent Tensors and their Hessian

Persistent tensors, introduced in [Quantum 8 (2024), 1238], and inspired by quantum information theory, form a recursively defined class of tensors that remain stable under the substitution method and thereby yield nontrivial lower bounds on tensor rank. In this work, we investigate the symmetric case-namely, symmetric persistent tensors, or equivalently, persistent polynomials. We establish that a symmetric tensor in $\mathrm{Sym}^n \mathbb{C}^d$ is persistent if the determinant of its Hessian equals the $d(n-2)$-th power of a nonzero linear form. The converse is verified for cubic tensors ($n=3$) or for $d \leq 3$, by leveraging classical results of B. Segre. Moreover, we demonstrate that the Hessian of a symmetric persistent tensor factors as the $d$-th power of a form of degree $(n-2)$. Our main results provide an explicit necessary and sufficient criterion for persistence, thereby offering an effective algebraic characterization of this class of tensors. Beyond characterization, we present normal forms in small dimensions, place persistent polynomials within prehomogeneous geometry, and connect them with semi-invariants, homaloidal polynomials, and Legendre transforms. Particularly, we prove that all persistent cubics are homaloidal.

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arXiv2025

The general ternary form can be recovered by its Hessian

The Hessian map is the rational map that sends a homogeneous polynomial to the determinant of its Hessian matrix. We prove that the Hessian map is birational on its image for ternary forms of degree $d\ge 4$, $d\neq 5$, by considering the action of the orthogonal group. In a previous paper we proved the analogous result for binary forms, with more geometric techniques.

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arXiv2024

Generalized identifiability of sums of squares

Let $f$ be a homogeneous polynomial of even degree $d$. We study the decompositions $f=\sum_{i=1}^r f_i^2$ where $\mathrm{deg} f_i=d/2$. The minimal number of summands $r$ is called the $2$-rank of $f$, so that the polynomials having $2$-rank equal to $1$ are exactly the squares. Such decompositions are never unique and they are divided into $\mathrm{O}(r)$-orbits, the problem becomes counting how many different $\mathrm{O}(r)$-orbits of decomposition exist. We say that $f$ is $\mathrm{O}(r)$-identifiable if there is a unique $\mathrm{O}(r)$-orbit. We give sufficient conditions for generic and specific $\mathrm{O}(r)$-identifiability. Moreover, we show the generic $\mathrm{O}(r)$-identifiability of ternary forms.

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arXiv2024

Effective methods for plane quartics, their theta characteristics and the Scorza map

This is a revised version of the lecture notes prepared for the workshop on "Plane quartics, Scorza map and related topics", held in Catania, January 19-21, 2016. The last section contains eight Macaulay2 scripts on theta characteristics and the Scorza map, with a tutorial. The first sections give an introduction to these scripts. The tutorial contains a list of the 36 Scorza preimages of the Edge quartic.

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arXiv2024

On the degree of varieties of sum of squares

We study the problem of how many different sums of squares decompositions a general polynomial $f$ with SOS-rank $k$ admits. We show that there is a link between the variety $\mathrm{SOS}_k(f)$ of all SOS-decompositions of $f$ and the orthogonal group $\mathrm{O}(k)$. We exploit this connection to obtain the dimension of $\mathrm{SOS}_k(f)$ and show that its degree is bounded from below by the degree of $\mathrm{O}(k)$. In particular, for $k=2$ we show that $\mathrm{SOS}_2(f)$ is isomorphic to $\mathrm{O}(2)$ and hence the degree bound becomes an equality. Moreover, we compute the dimension of the space of polynomials of SOS-rank $k$ and obtain the degree in the special case $k=2$.

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arXiv2024

Vector bundles without intermediate cohomology and the trichotomy result

Horrocks proved in 1964 that vector bundles on $P^n$ without intermediate cohomology split as direct sum of line bundles. This result has been the starting point of a great research activity on other varieties, showing interesting connections with derived categories and other areas. We follow some paths into this fascinating story, which has classical roots. The story has a culmination with the trichotomy result (finite/tame/wild) for arithmetically Cohen Macaulay (ACM) varieties obtained by Faenzi and Pons-Llopis in 2021. This is an expanded version of the talk given at the conference "Homemade Algebraic Geometry" in July 2023 at Alcalá de Henares celebrating Enrique Arrondo's 60th birthday.

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arXiv2023

Taylor Polynomials of Rational Functions

A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed. In one variable, Taylor varieties are given by rank constraints on Hankel matrices. Inversion of the natural parametrization is known as Padé approximation. We study the dimension and defining ideals of Taylor varieties. Taylor hypersurfaces are interesting for projective geometry, since their Hessians tend to vanish. In three and more variables, there exist defective Taylor varieties whose dimension is smaller than the number of parameters. We explain this with Fröberg's Conjecture in commutative algebra.

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arXiv2022

Tensor Rank and Complexity

These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will be expressed in terms of tensor (border) rank, (border) symmetric rank and the asymptotic rank of certain tensors. We introduce the multilinear rank of a tensor as well, deal with the concept of tensor equivalence and study prehomogeneous vector spaces with the castling transform. Moreover, we treat Apolarity Theory and use it to determine the symmetric rank (Waring rank) of some symmetric tensors.

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arXiv2021

Asymptotics of degrees and ED degrees of Segre products

Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this paper, we study the asymptotic behavior of these two degrees of some Segre products and their dual varieties. We analyze the asymptotics of degrees of (hypercubical) hyperdeterminants, the dual hypersurfaces to Segre varieties. We offer an alternative viewpoint on the stabilization of the ED degree of some Segre varieties. Although this phenomenon was incidentally known from Friedland-Ottaviani's formula expressing the number of singular vector tuples of a general tensor, our approach provides a geometric explanation. Finally, we establish the stabilization of the degree of the dual variety of a Segre product $X\times Q_{n}$, where $X$ is a projective variety and $Q_n\subset \mathbb{P}^{n+1}$ is a smooth quadric hypersurface.

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arXiv2021

The critical space for orthogonally invariant varieties

Let $q$ be a nondegenerate quadratic form on $V$. Let $X\subset V$ be invariant for the action of a Lie group $G$ contained in $SO(V,q)$. For any $f\in V$ consider the function $d_f$ from $X$ to $C$ defined by $d_f(x)=q(f-x)$. We show that the critical points of $d_f$ lie in the subspace orthogonal to ${\mathfrak g}\cdot f$, that we call critical space. In particular any closest point to $f$ in $X$ lie in the critical space. This construction applies to singular t-ples for tensors and to flag varieties and generalizes a previous result of Draisma, Tocino and the author. As an application, we compute the Euclidean Distance degree of a complete flag variety.

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arXiv2021

A footnote to a footnote to a paper of B. Segre

The paper is devoted to a detailed study of sextics in three variables having a decomposition as a sum of nine powers of linear forms. This is the unique case of a Veronese image of the plane which, in the terminology introduced by Ciliberto and the first author in [12], is weakly defective, and non-identifiable. The title originates from a paper of 1981, where Arbarello and Cornalba state and prove a result on plane curves with preassigned singularities, which is relevant to extend the studies of B. Segre on special linear series on curves. We explore the apolar ideal of a sextic $F$ and the associated catalecticant maps, in order to determine the minimal decompositions. A particular attention is played to the postulation of the decompositions. Starting with forms with a decomposition $A$ of length $9$, the postulation of $A$ determines several loci in the $9$-secant of the $6$-Veronese image of $\mathbb P^2$, which include the lower secant varieties, and the ramification locus, where the decomposition is unique. We prove that equations of all these loci, including the $8$-th and the $7$-th secant varieties, are provided by minors of the catalecticant maps and by the invariant $H_{27}$ that we describe in Section 4.

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arXiv2020

Tensors with eigenvectors in a given subspace

The first author with B. Sturmfels studied the variety of matrices with eigenvectors in a given linear subspace, called Kalman variety. We extend that study from matrices to symmetric tensors, proving in the tensor setting the irreducibility of the Kalman variety and computing its codimension and degree. Furthermore we consider the Kalman variety of tensors having singular t-ples with the first component in a given linear subspace and we prove analogous results, which are new even in the case of matrices. Main techniques come from Algebraic Geometry, using Chern classes for enumerative computations.

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arXiv2020

Fine-Structure Classification of Multiqubit Entanglement by Algebraic Geometry

We present a fine-structure entanglement classification under stochastic local operation and classical communication (SLOCC) for multiqubit pure states. To this end, we employ specific algebraic-geometry tools that are SLOCC invariants, secant varieties, to show that for $n$-qubit systems there are $\lceil\frac{2^{n}}{n+1}\rceil$ entanglement families. By using another invariant, $\ell$-multilinear ranks, each family can be further split into a finite number of subfamilies. Not only does this method facilitate the classification of multipartite entanglement, but it also turns out to be operationally meaningful as it quantifies entanglement as a resource.

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arXiv2020

The Distance Function from a Real Algebraic Variety

For any (real) algebraic variety $X$ in a Euclidean space $V$ endowed with a nondegenerate quadratic form $q$, we introduce a polynomial $\mathrm{EDpoly}_{X,u}(t^2)$ which, for any $u\in V$, has among its roots the distance from $u$ to $X$. The degree of $\mathrm{EDpoly}_{X,u}$ is the {\em Euclidean Distance degree} of $X$. We prove a duality property when $X$ is a projective variety, namely $\mathrm{EDpoly}_{X,u}(t^2)=\mathrm{EDpoly}_{X^\vee,u}(q(u)-t^2)$ where $X^\vee$ is the dual variety of $X$. When $X$ is transversal to the isotropic quadric $Q$, we prove that the ED polynomial of $X$ is monic and the zero locus of its lower term is $X\cup(X^\vee\cap Q)^\vee$.

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arXiv2020

The Hessian map

In this paper we study the Hessian map $h_{d,r}$ which associates to any hypersurface of degree $d$ in ${\mathbb P}^r$ its Hessian hypersurface. We study general properties of this map and we prove that: $h_{d,1}$ is birational onto its image if $d\geq 5$; we study in detail the maps $h_{3,1}$, $h_{4,1}$ and $h_{3,2}$; we study the restriction of the Hessian map to the locus of hypersurfaces of degree $d$ with Waring rank $r+2$ in ${\mathbb P}^r$, proving that this restriction is injective as soon as $r\geq 2$ and $d\geq 3$, which implies that $h_{3,3}$ is birational onto its image; we prove that the differential of the Hessian map is of maximal rank on the generic hypersurfaces of degree $d$ with Waring rank $r+2$ in ${\mathbb P}^r$, as soon as $r\geq 2$ and $d\geq 3$.

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arXiv2018

Best rank-$k$ approximations for tensors: generalizing Eckart-Young

Given a tensor $f$ in a Euclidean tensor space, we are interested in the critical points of the distance function from $f$ to the set of tensors of rank at most $k$, which we call the critical rank-at-most-$k$ tensors for $f$. When $f$ is a matrix, the critical rank-one matrices for $f$ correspond to the singular pairs of $f$. The critical rank-one tensors for $f$ lie in a linear subspace $H_f$, the critical space of $f$. Our main result is that, for any $k$, the critical rank-at-most-$k$ tensors for a sufficiently general $f$ also lie in the critical space $H_f$. This is the part of Eckart-Young Theorem that generalizes from matrices to tensors. Moreover, we show that when the tensor format satisfies the triangle inequalities, the critical space $H_f$ is spanned by the complex critical rank-one tensors. Since $f$ itself belongs to $H_f$, we deduce that also $f$ itself is a linear combination of its critical rank-one tensors.

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