Cristiano Bocci

• University of Siena

In this paper, we generalize and study the concept of Hadamard product of ideals of projective varieties to the case of monomial ideals. We have a research direction similar to the one of the join of monomial ideals contained in a paper of Sturmfels and Sullivant. In the second part of the paper, we give a brief tutorial on the Hadamard.m2 package...

In this chapter we investigate the Hadamard products of degenerate subvarieties \(X_i\subset \mathbb {P}^n\), that is each variety \(X_i\) lies in some hyperplane. In particular, we want to determine the dimension and the degree of the Hadamard product of these varieties using the dimensions and the degrees of the varieties \(X_i\).

In this chapter, we start our journey to discover the Hadamard product of projective varieties. We first introduce two different definitions of Hadamard products, and then we describe the main tools, such as Hadamard transformations, and some basic results, such as Hadamard–Terracini Lemma, that we will use in the whole book. In this chapter, we al...

In this chapter, we will describe some connections among the Hilbert functions of X, \(X'\) and \(X\star X'\). However, a complete description of these connections is not known yet. In this chapter, we focus our attention on linear spaces, collinear points, degenerate varieties, and binomial varieties.

In Chap. 9, we saw that it is possible to construct star configurations via Hadamard products. With this construction we have a complete control both on the coordinates of the points forming the star configuration and on the equations of the hyperplanes involved. Thus, the question if other interesting geometrical objects can be obtained via Hadama...

The natural generalization of the notion of binomial hypersurface of the previous chapter is the one of binomial variety: that is, a variety whose associated ideal is a binomial ideal. In this chapter we study binomial varieties and their Hadamard products.

In the previous chapter, we studied the behaviour of the Hadamard product of points and lines without any genericity assumption. We will now investigate the Hadamard product \(X\star X'\) of two sets of collinear points X and \(X'\) lying on two lines L and \(L'\). Thus we will combine the genericity and the non-genericity.

The Hadamard product of two hypersurfaces is, in general, the whole ambient space. However, there are cases in which this product is again a hypersurface.

In this chapter, we give a relevant application of the Hadamard powers of a line: Theorem 9.1shows how to naturally construct star configurations using Hadamard products. The interest in star configurations has recently increased because of their extremal behaviour with respect to many open problems such as, for example, symbolic powers of ideals.

In this chapter, we consider for the first time geometrical objects not necessarily in general position. We start with zero-dimensional schemes and linear spaces: Points can have many zero coordinates, and linear spaces can intersect coordinate hyperplanes in dimension greater than the expected one. These facts force us to study all possible pathol...

In this chapter, we start to study the behaviour of linear varieties under Hadamard products. We begin with some useful results about the Hadamard multiplication of linear spaces by a point. Then, we study Hadamard products, and powers, of lines. Successively, we consider Hadamard products of linear spaces. Using Tropical Geometry, we are able to g...

In this chapter, we introduce some open problems about Hadamard products of projective varieties and related topics. Since this is an active and young area of research, we hope that this chapter will give a chance to the interested reader to actively get involved in the subject.

In the previous chapters, we mainly use Definition 1.5 and Lemma 1.1, to compute several instances of Hadamard products of projective varieties. In order to do this, we used radical ideals associated to the varieties, but Definition 1.5 allows us to use different families of ideals, such as monomial ideals, square-free binomial ideals, and edge ide...

In this paper we address the question if, for points $$P, Q \in \mathbb {P}^{2}$$ P , Q ∈ P 2 , $$I(P)^{m} \star I(Q)^{n}=I(P \star Q)^{m+n-1}$$ I ( P ) m ⋆ I ( Q ) n = I ( P ⋆ Q ) m + n - 1 and we obtain different results according to the number of zero coordinates in P and Q . Successively, we use our results to define the so called Hadamard fat...

In this paper we address the question if, for points $P, Q \in \mathbb{P}^{2}$, $I(P)^{m} \star I(Q)^{n}=I(P \star Q)^{m+n-1}$ and we obtain different results according to the number of zero coordinates in $P$ and $Q$. Successively, we use our results to define the so called Hadamard fat grids, which are the result of the Hadamard product of two se...

The \(\mathsf {NP}\)-complete graph problem Cluster Editing seeks to transform a static graph into disjoint union of cliques by making the fewest possible edits to the edge set. We introduce a natural interpretation of this problem in the setting of temporal graphs, whose edge-sets are subject to discrete changes over time, which we call Editing to...

We show how to construct a stick figure of lines in $${\mathbb {P}}^3$$ P 3 using the Hadamard product of projective varieties. Then, applying the results of Migliore and Nagel, we use such a stick figure to build a Gorenstein set of points with given $$h-$$ h - vector $${\varvec{h}}$$ h . Since the Hadamard product is a coordinate-wise product, we...

In this paper we characterize hypersurfaces for which their Hadamard product is still a hypersurface Then we study hypersurfaces and, more generally, varieties which are indempotent under Hadamard powers.

The NP-complete graph problem Cluster Editing seeks to transform a static graph into disjoint union of cliques by making the fewest possible edits to the edge set. We introduce a natural interpretation of this problem in the setting of temporal graphs, whose edge-sets are subject to discrete changes over time, which we call Editing to Temporal Cliq...

We show how to construct a stick figure of lines in $\mathbb{P}^3$ using the Hadamard product of projective varieties. Then, applying the results of Migliore and Nagel, we use such stick figure to build a Gorenstein set of points with given $h-$vector ${\mathbf h}$. Since the Hadamard product is a coordinate-wise product, we show, at the end, how t...

In this paper we, first, characterize hypersurfaces for which their Hadamard product is still a hypersurface. Then we pass to study hypersurfaces and, more generally, varieties which are idempotent under Hadamard powers.

Given a distance matrix $D$, we study the behavior of its compaction vector and reduction matrix with respect to the problem of the realization of $D$ by a weighted graph. To this end, we first give a general result on realization by $n-$cycles and successively we mainly focus on graphs of genus 1, presenting an algorithm which determines when a di...

We introduce the notion of confinement of decompositions for forms or vector of forms. The confinement, when it holds, lowers the number of parameters that one needs to consider, in order to find all the possible decompositions of a given set of data. With the technique of confinement, we obtain here two results. First, we give a new, shorter proof...

We introduce the notion of confinement of decompositions for forms or vector of forms. The confinement, when it holds, lowers the number of parameters that one needs to consider, in order to find all the possible decompositions of a given set of data. With the technique of confinement, we obtain here two results. First, we give a new, short proof o...

In this paper, the authors propose a novel methodology to measure the volume of masses of granular material by means of wired or wireless sensor networks. The proposed technological framework exploits a grid of sensor nodes, each of them in charge of measuring the level of a layer of granular material in a specific point. Since this layout allows t...

We collect in this chapter some of the most useful operations on tensors, in view of the applications to Algebraic Statistics.

In this chapter, we introduce the concept of model, essential point of statistical inference. The concept is reviewed here by our algebraic interpretation.

No, we are not exaggerating. We are instead simplifying. In fact, many of the phenomena associated with the main statistic models are better understood if studied, at least in the first measure, from the projective point of view and on an algebraically closed numerical field. The main link between Algebraic Statistics and Projective Algebraic Geome...

The scope of this part of the book is to provide a quick introduction to the main tools of the algebraic geometry of projective spaces that are necessary to understand some aspects of algebraic models in Statistics.

In this chapter, we make a specific analysis of the behavior of symmetric tensors, with respect to the rank and the decomposition. We will see, indeed, that besides their utility to understand some models of random systems, symmetric tensors have a relevant role in the study of the algebra and the computational complexity of polynomials.

Groebner bases represent the most powerful tool for computational algebra, in particular for the study of polynomial ideals. In this chapter, based on [1, Chap. 2], we give a brief survey on the subject. For a deeper study of it, we suggest [1, 2].

This section contains the basic definitions with which we will construct our statistical theory. It is important to point out right away that in the field of Algebraic Statistics, a still rapidly developing area of study, the basic definitions are not yet standardized. Therefore, the definitions which we shall use in this text can differ significan...

An intermediate case between total independence and generic situations of the dependence of random variables concerns the so-called conditional independence.

The main objects of multi-linear algebra that we will use in the study of Algebraic Statistics are multidimensional matrices, that we will call tensors.

The study of the rank of tensors has a natural geometric counterpart in the study of secant varieties. Secant varieties or, more generally, joins are a relevant object for several researches on the properties of projective varieties.

The concept of dimension of a projective variety is a fairly intuitive but surprisingly delicate invariant, from an algebraic point of view.

The chapter contains the proof of the Chow’s Theorem, a fundamental result for algebraic varieties with an important consequence for the study of statistical models. It states that, over an algebraically closed field, like \(\mathbb C\), the image of a projective (or multiprojective) variety X under a projective map is a Zariski closed subset of th...

In this chapter, we focus on some basic examples in Probability and Statistics. We phrase these concepts using the language and definitions we have given in the previous chapter.

This book provides an introduction to various aspects of Algebraic Statistics with the principal aim of supporting Master’s and PhD students who wish to explore the algebraic point of view regarding recent developments in Statistics. The focus is on the background needed to explore the connections among discrete random variables. The main objects t...

We study three different quasi-symmetry models and three different mixture models of $n\times n\times n$ tensors for modeling rater agreement data. For these models we give a geometric description of the associated varieties and we study their invariants distinguishing between the case $n=2$ and the case $n>2$. Finally, for the two models for pairw...

In this paper we address the Hadamard product of not necessarily generic linear varieties, looking in particular at its Hilbert function. We find that the Hilbert function of the Hadamard product \(X\star Y\) of two varieties, with \(\dim (X), \dim (Y)\le 1\), is the product of the Hilbert functions of the original varieties X and Y. Moreover, the...

In this work we define log-linear models to compare several square contingency tables under the quasi-independence or the quasi-symmetry model, and the relevant Markov bases are theoretically characterized. Through Markov bases, an exact test to evaluate if two or more tables fit a common model is introduced. Two real-data examples illustrate the u...

In this work we define log-linear models to compare several square contingency tables under the quasi-independence or the quasi-symmetry model, and the relevant Markov bases are theoretically characterized. Through Markov bases, an exact test to evaluate if two or more tables fit a common model is introduced. Two real-data examples illustrate the u...

Given a squarefree monomial ideal $I \subseteq R =k[x_1,\ldots,x_n]$, we show that $\widehat\alpha(I)$, the Waldschmidt constant of $I$, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of $I$. By applying results from fractional graph theory, we can then express $\widehat\alpha(I)$ in terms of...

Let $T$ be a real tensor of (real) rank $r$. $T$ is 'identifiable' when it has a unique decomposition in terms of rank $1$ tensors. There are cases in which the identifiability fails over the complex field, for general tensors of rank $r$. This behavior is quite peculiar when the rank $r$ is submaximal. Often, the failure is due to the existence of...

Let $T$ be a real tensor of (real) rank $r$. $T$ is 'identifiable' when it has a unique decomposition in terms of rank $1$ tensors. There are cases in which the identifiability fails over the complex field, for general tensors of rank $r$. This behavior is quite peculiar when the rank $r$ is submaximal. Often, the failure is due to the existence of...

In this paper, we address the Hadamard product of linear varieties not necessarily in general position. In ℙ2, we obtain a complete description of the possible outcomes. In particular, in the case of two disjoint finite sets X and X′ of collinear points, we get conditions for X ⋆ X′ to be either a collinear finite set of points or a grid of |X||X′|...

In this paper we address the Hadamard product of linear varieties not necessarily in general position. In $\mathbb{P}^2$ we obtain a complete description of the possible outcomes. In particular, in the case of two disjoint finite sets X and X' of collinear points, we get conditions for their hadamard product to be either a collinear finite set of p...

It is a fundamental challenge for many problems of significant current interest in algebraic geometry and commutative algebra to understand symbolic powers $I^{(m)}$ of homogeneous ideals $I$ in polynomial rings, particularly ideals of linear varieties. Such problems include computing Waring ranks of polynomials, determining the occurrence of equal...

We describe properties of Hadamard products of algebraic varieties. We show any Hadamard power of a line is a linear space, and we construct star configurations from products of collinear points. Tropical geometry is used to find the degree of Hadamard products of other linear spaces.

We study the symbolic powers of the Stanley–Reisner ideal IBn of a bipyramid Bn over a n-gon Qn. Using a combinatorial approach, based on analysis of subtrees in Qn we compute the Waldschmidt constant of IBn.

We prove that the general tensor of size \(2^n\) and rank \(k\) has a unique decomposition as the sum of decomposable tensors if \(k\le 0.9997\frac{2^n}{n+1}\) (the constant 1 being the optimal value). Similarly, the general tensor of size \(3^n\) and rank \(k\) has a unique decomposition as the sum of decomposable tensors if \(k\le 0.998\frac{3^n}...

In this paper, we analyze the tropical decomposition of the weighted adjacency matrix of an alignment graph in terms of shifted diagonal matrices. We use this decomposition to describe a way to compute the minimal cost of an alignment applying a refined version of the Floyd-Warshall algorithm.

In the week 3--9, October 2010, the Mathematisches Forschungsinstitut at Oberwolfach hosted a mini workshop Linear Series on Algebraic Varieties. These notes contain a variety of interesting problems which motivated the participants prior to the event, and examples, results and further problems which grew out of discussions during and shortly after...

Given an undirected graph $G$, we define a new object $H_G$, called the mp-chart of $G$, in the max-plus algebra. We use it, together with the max-plus permanent, to describe the complexity of graphs. We show how to compute the mean and the variance of $H_G$ in terms of the adjacency matrix of $G$ and we give a central limit theorem for $H_G$. Fina...

In this paper we study how perturbing a matrix changes its nonnegative rank. We prove that the nonnegative rank can only increase in a neighborhood of a matrix with no zero columns. Also, we describe some special families of perturbations. We show how our results relate to statistics in terms of the study of maximum likelihood estimation for mixtur...

Guided by evidence coming from a few key examples and attempting to unify previous work of Chudnovsky, Esnault-Viehweg, Eisenbud-Mazur, Ein-Lazarsfeld-Smith, Hochster-Huneke and Bocci-Harbourne, Harbourne and Huneke recently formulated a series of conjectures that relate symbolic and regular powers of ideals of fat points in ${\bf P}^N$. In this pa...

We prove that a product of $m>5$ copies of $\PP^1$, embedded in the projective space $\PP^r$ by the standard Segre embedding, is $k$-identifiable (i.e. a general point of the secant variety $S^k(X)$ is contained in only one $(k+1)$-secant $k$-space), for all $k$ such that $k+1\leq 2^{m-1}/m$.

We classify sets Z of points in the projective plane, for which the difference between the minimal degrees of curves containing 2Z and Z respectively, is small. MSC. 14N05

Starting from the new edition, published in 2010, of G. V. Schiaparelli's paper "Studio comparativo tra le forme organiche naturali e le forme geometriche pure" (1898), we show the theorical and mathematical developments inspired by such work. Our aim is to propose both a geometrical and dynamical models and related numerical simulations.

In this paper we study how perturbing a matrix changes its non-negative rank. We prove that the non-negative rank is upper-semicontinuos and we describe some special families of perturbations. We show how our results relate to Statistics in terms of the study of Maximum Likelihood Estimation for mixture models.

Computer models and computer simulations are crucial for understanding complex phenomena because they compel the explicit enumeration of all variables and the exact specification of all relations between them. In this paper we discuss a computer model for a phenotypical theory of evolution which, in our opinion, is well suited to simulate the compl...

Here we introduce the concept of special effect curve which permits to study, from a different point of view, special linear systems in P^2, i.e. linear system with general multiple base points whose effective dimension is strictly greater than the expected one. In particular we study two different kinds of special effect: the \alpha-special effect...

In this work we study several types of diagonal-effect models for two-way contingency tables in the framework of Algebraic Statistics. We use both toric models and mixture models to encode the different behavior of the diagonal cells. We compute the invariants of these models and we explore their geometrical structure. Comment: 20 pages

Given a symbolic power of a homogeneous ideal in a polynomial ring, we study the problem of determining which powers of the ideal contain it. For ideals defining 0-dimensional subschemes of projective space, as an immediate corollary of our previous paper (arXiv:0706.3707) we give a complete solution in terms of the least degrees of nonzero element...

Let T be a weighted tree with n numbered leaves and let D be its distance matrix, so D(i,j) is the distance between the leaves i and j. If m is an integer between 2 and n, we prove a tropical formula to compute the m-dissimilarity map of T (i.e. the weights of the subtrees of T with m leaves), given D. For m equal to 3, we present a tropical descri...

We classify all sequences of integers that can be, up to a shift, the cohomology sequence {h 1 (E(n))} of a rank 2 bundle E on ℙ 2. We show how some of the main invariants of the bundle can be read from the sequence.

In this paper we present an historical reconstruction and analysis of theoretic developments which, in the context of the school of Peano, led from H. Grassmann's approach to the realization of vector calculus and the theory of homographies. Our aim is also to attempt a generalization of the fundamental ideas introduced by Peano (and Grassmann). In...

In this expository work we describe the main aspects of Phyloge-netic Algebraic Geometry. In particular, we will focus our attention on the technique of flattening of a n−dimensional tensor. Our interest in flattening is due to the fact that they are strictly related with the study of secant varieties of Segre varieties.

We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results on the structure of the set of pairs $(r,m)$ such that $I^{(m)}\subseteq I^r$. As corollaries, we show that $I^...

In this expository work we describe the main aspects of the so-called Phylogenetic Algebraic Geometry which concerns with the study of algebraic varieties representing statistical models of evolution. In particular, we mainly describe how this field of research can be used to infer phylogenies.

In this paper we present a geometric model for a proposal of axiomatization of Evolution Theory. For this aim, we use suitable tools of Geometry and Topology. In particular, we define the concept of fertility factor as a main instrument for the studying of speciation. This concept, in our opinion, has an important biological meaning.

In this paper, we present a method to inductively construct Gorenstein ideals of any codimension c. We start from a Gorenstein ideal I of codimension c contained in a complete intersection ideal J of the same codimension, and we prove that under suitable hypotheses there exists a new Gorenstein ideal contained in the residual ideal I : J. We compar...

Here we introduce the concept of special effect varieties in higher dimension and we generalize to the n-dimensional projective space, n>=3, the two conjectures given in AG/0410527 for the planar case. Finally, we propose some examples on the product of projective spaces and we show how these results fit with the ones of Catalisano, Geramita and Gi...

Here we study zero-dimensional subschemes of ruled varieties, mainly Hirzebruch surfaces and rational normal scrolls, by applying the Horace method and the Terracini method

We generalize the classical Terracini’s Lemma to higher order osculating spaces to secant varieties. As an application, we address with the so-called Horace method the case of thed-Veronese embedding of the projective 3-space.

These are notes of the lectures given by the authors during the school/workshop "Polynomial Interpolation and Projective Embeddings". We mainly focus our attention on the planar case and on the Segre and Harbourne-Hirschowitz Conjectures. We discuss the state of the art introducing several results and different techniques.

After the structure theorem of Buchsbaum and Eisenbud on Gorenstein ideals of codimension 3, much progress was made in this area from the algebraic point of view; in particular some characterizations of these ideals using h-vectors and minimal resolutions were given. On the other hand, the liaison theory gives some tools to exploit, but, at the sam...

We introduce the notion of "sub–defective' varieties as those varieties X for which a general tangent space intersects "too many" other tangent spaces to X. We show that this notion has an unexpected link with the study of varieties with degenerate osculating behavior. Namely we show that sub-defective surfaces are precisely those surfaces whose ge...