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Luca Chiantini

Publications and source records attributed to Luca Chiantini.

At least 19 recordsLinked to original sources
arXiv2026

Artin-Schreier geproci configurations in projective spaces of arbitrary dimension

We construct finite geproci sets in every projective dimension and in every positive characteristic by introducing $\mathbb{F}_N$-Artin-Schreier configurations. In $\mathbb{P}^3$, we characterize exactly when such configurations are geproci: an $\mathbb{F}_N$-Artin-Schreier configuration on $q$ lines spanning $\mathbb{P}^3$ is $(q,N)$-geproci if and only if $q\leq N$. We then develop a lifting construction which produces geproci sets in $\mathbb{P}^n$ for every $n\geq 3$.

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arXiv2026

Enumerative geometry of skew lines in $\mathbb P^3$ with a given associated finite group

For any finite set $\mathcal L$ of 3 or more skew lines in $\mathbb P^3_{\overline{K}}$ over an algebraically closed field $\overline{K}$ of arbitrary characteristic, there is a canonical associated subgroup $G_{\mathcal L}$ of ${\rm PGL}_2(\overline{K})$. Given a finite subgroup $G\subset{\rm PGL}_2(\overline{K})$ we study which configurations of lines have $G_{\mathcal L}=G$. We derive an upper bound on the number $|\mathcal L|$ of lines in terms of the order $|G|$ of the group $G$ and as an application we classify up to projective equivalence which sets $\mathcal L$ in $\mathbb P^3_{\mathbb C}$ have $G_{\mathcal L}=G$ for certain finite nonabelian groups $G$.

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arXiv2026

Weddle schemes

The classical Weddle surface is the locus of vertices of quadric cones through six points in $\mathbb{P}^3$ in linear general position. Equivalently, it is the closure of the locus of centers of projection from which those six points map to six points on a plane conic. Motivated by this 1850 construction of T. Weddle, we introduce $d$-Weddle schemes for finite point sets $Z\subset \mathbb{P}^n$, defined by an analogous projection-to-degree-$d$ condition. Our main tool is Macaulay duality, which yields a natural multiplication map in an Artinian algebra defined by powers of linear forms. This viewpoint connects $d$-Weddle schemes to unexpected cones and interprets them as non-Lefschetz loci for these multiplication maps. Parallel to this, we give an analysis from the point of view of interpolation matrices, and we explain the connections between these approaches. For a general set $Z\subset \mathbb{P}^n$ of $\binom{d+n}{n}$ points, we show that the $d$-Weddle scheme is a hypersurface and we compute its degree. We also study general sets whose cardinalities are "near" such a binomial coefficient, where the Weddle scheme has higher codimension. Returning to sets of six points (not always in linear general position), we discuss special configurations in which the appropriate Weddle scheme is reducible, or even nonreduced.

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arXiv2025

Waring decompositions of special binomials

We determine the Waring rank of homogeneous polynomials of the form $x^ky^kz^k + \ell^{3k}$ where $\ell$ is a linear form. The result is based on the study of the Hilbert function and the resolution of special configurations of points in $\mathbb{P}^2$. As a byproduct of our result, we show that the monomial $x^ky^kz^k$ does not have irredundant decompositions of length $(k+1)^2 +1$.

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arXiv2025

Intersection of curves in projective 4 space

Given two distinct reduced, irreducible curves of given degrees, contained in projective space but whose union is not contained in a hyperplane, what is the largest number of points of intersection they can have? When the projective space is the plane, this is trivial. For projective 3 space this problem was solved independently by Diaz and by Giuffrida in 1986. They showed that two curves achieving the maximum number of intersection points have to be rational curves on a smooth surface of minimal degree, i.e., a quadric surface. Note that these curves are far from being arithmetically Cohen-Macaulay. In contrast, Hartshorne and Mir\'o-Roig addressed this problem in 2015 for space curves under the assumption that the curves are arithmetically Cohen-Macaulay (ACM), introducing very deep techniques and obtaining very different results from Diaz and Giuffrida. Diaz and Giuffrida also gave initial results in dimensions greater than 3. Here we continue this study for dimension 4. We introduce a number B defined in terms of the degrees of the curves and prove that when both curves lie on a surface of minimal degree (thus a cubic surface) then the number of points of intersection is at most B. Moreover, we conjecture that B is always an upper bound and we prove this conjecture in many cases, including when at least one of the curves is ACM. Our approach focuses on the genera of the curves and their union. In addition we define a second number B' in terms of the degrees and the genus of the union which we can show bounds the number of points of intersection above, and we use a variety of methods to study how B and B' compare.

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arXiv2025

Weddle loci of linear systems of quadrics and the rank of partially symmetric tensors

We establish a connection between properties of partially symmetric tensors (i.e. tensors associated to linear systems of quadric hypersurfaces) and the geometry of some related loci, generalization of the Weddle loci introduced in \cite{CFF+22} for their role in the study of configurations of points and interpolation problems. In particular, we consider linear systems of plane conics and linear systems of quadric surfaces, and show that when the associated tensors have low rank, then the singularities of the corresponding Weddle loci satisfy a (sharp) lower bound. Thus, we obtain a criterion to exclude that the rank of some partially symmetric tensors is too low. In the final section, devoted to partially symmetric $n\times n\times n$ tensors which lie in one component $M$ of a standard decomposition of the space of $3$-dimensional tensors (\cite{IR22}), we prove that the number of singular points of the Weddle locus associated to a general tensor in $M$ equals the (recursively defined) $n$-th Jacobsthal number.

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arXiv2024

Finite sets of points in $\mathbb{P}^4$ with special projection properties

In this note we introduce the notion of $(b,d)$-geprofi sets and study their basic properties. These are sets of $bd$ points in $\mathbb{P}^4$ whose projection from a general point to a hyperplane is a full intersection, i.e., the intersection of a curve of degree $b$ and a surface of degree $d$. We show that such nontrivial sets exist if and only if $b\geq 4$ and $d\geq 2$. Somewhat surprisingly, for infinitely many values of $b$ and $d$ there exist such sets in linear general position. The note contains open questions and problems.

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arXiv2024

Product of Tensors and Description of Networks

Any kind of network can be naturally represented by a Directed Acyclic Graph (DAG); additionally, activation functions can determine the reaction of each node of the network with respect to the signal(s) incoming. We study the characterization of the signal distribution in a network under the lens of tensor algebra. More specifically, we describe every activation function as tensor distributions with respect to the nodes, called \textit{activation tensors}. The distribution of the signal is encoded in the \textit{total tensor} of the network. We formally prove that the total tensor can be obtained by computing the \textit{Batthacharya-Mesner Product} (BMP), an $n$-ary operation for tensors of order $n$, on the set of the activation tensors properly ordered and processed via two basic operations, that we call \textit{blow} and \textit{forget}. Our theoretical framework can be validated through the related code developed in Python.

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arXiv2023

Geproci sets on skew lines in $\mathbb P^3$ with two transversals

The purpose of this work is to pursue classification of geproci sets. Specifically we classify $[m,n]$-geproci sets which consist of $m=4$ points on each of $n$ skew lines, assuming the skew lines have two transversals in common. We show that in this case $n\leq 6$. Moreover we show that all geproci sets of this type are contained in the \emph{standard construction} for $m=4$ introduced in arXiv:2209.04820. Finally, we propose a conjectural representation for all geproci sets of this type, irrespective of the number $m$ of points on each skew line.

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arXiv2023

Combinatorics of skew lines in $\mathbb P^3$ with an application to algebraic geometry

This article introduces a previously unrecognized combinatorial structure underlying configurations of skew lines in $\mathbb{P}^3$, and reveals its deep and surprising connection to the algebro-geometric concept of geproci sets. Given any field $\mathbb{K}$ and a finite set $\mathcal L$ of 3 or more skew lines in $\mathbb{P}^3_\mathbb{K}$, we associate to it a group $G_{\mathcal L}$ and a groupoid $C_{\mathcal L}$ whose action on the union $\cup_{L\in\mathcal L}L$ provides orbits which have a rich combinatorial structure. We characterize when $G_{\mathcal L}$ is abelian and give partial results on its finiteness. The notion of \emph{collinearly complete} subsets is introduced and shown to correspond exactly to unions of groupoid orbits. In the case where $\mathbb{K}$ is a finite field and $\mathcal L$ is a full spread in $\mathbb{P}^3_\mathbb{K}$ (i.e., every point of $\mathbb{P}^3_\mathbb{K}$ lies on a line in $\mathcal{L}$), we prove that $G_{\mathcal L}$ being abelian characterizes the classical spread given by the fibers of the Hopf fibration. Over any algebraically closed field, we establish that finite unions of $C_{\mathcal L}$-orbits are geproci sets - that is, finite sets whose general projections to a plane are complete intersections. Furthermore, we prove a converse: if $\mathbb{K}$ is algebraically closed and $Z \subset \mathbb{P}^3_\mathbb{K}$ is a geproci set consisting of $m$ points on each of $s \geq 3$ skew lines $\mathcal L$ where the general projection of $Z$ is a complete intersection of type $(m, s)$, then $Z$ is a finite union of orbits of $C_{\mathcal L}$. This work thus uncovers a profound combinatorial framework governing geproci sets, providing a new bridge between incidence combinatorics and algebraic geometry.

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arXiv2023

Terracini loci of curves

We study subsets S of curves X whose double structure does not impose independent conditions to a linear series L, but there are divisors D in |L| singular at all points of S. These subsets form the Terracini loci of X. We investigate Terracini loci, with a special look towards their non-emptiness, mainly in the case of canonical curves, and in the case of space curves.

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arXiv2023

On the classification of certain geproci sets

In this short note we develop new methods toward the ultimate goal of classifying geproci sets in $\mathbb P^3$. We apply these methods to show that among sets of $16$ points distributed evenly on $4$ skew lines, up to projective equivalence there are only two distinct geproci sets. We give different geometric distinctions between these sets. The methods we develop here can be applied in a more general set-up; this is the context of the follow-up work arXiv:2308.00761.

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arXiv2023

Decompositions and Terracini loci of cubic forms of low rank

We study Waring rank decompositions for cubic forms of rank $n+2$ in $n+1$ variables. In this setting, we prove that if a concise form has more than one non-redundant decomposition of length $n+2$, then all such decompositions share at least $n-3$ elements, and the remaining elements lie in a special configuration. Following this result, we give a detailed description of the $(n+2)$-th Terracini locus of the third Veronese embedding of $n$-dimensional projective space.

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arXiv2022

Configurations of points in projective space and their projections

We call a set of points $Z\subset{\mathbb P}^{3}_{\mathbb C}$ an $(a,b)$-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point $P$ to a plane is a complete intersection of curves of degrees $a$ and $b$. Examples which we call grids have been known since 2011. The only nongrid nondegenerate examples previously known had $ab=12, 16, 20, 24, 30, 36, 42, 48, 54$ or $60$. Here, for any $4 \leq a \leq b$, we construct nongrid nondegenerate $(a,b)$-geproci sets in a systematic way. We also show that the only such example with $a=3$ is a $(3,4)$-geproci set coming from the $D_4$ root system, and we describe the $D_4$ configuration in detail. We also consider the question of the equivalence (in various senses) of geproci sets, as well as which sets occur over the reals, and which cannot. We identify several additional examples of geproci sets with interesting properties. We also explore the relation between unexpected cones and geproci sets and introduce the notion of $d$-Weddle schemes arising from special projections of finite sets of points. This work initiates the exploration of new perspectives on classical areas of geometry. We formulate and discuss a range of open problems in the final chapter.

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arXiv2022

On the study of decompositions of forms in four variables

In the space of sextic forms in 4 variables with a decomposition of length 18 we determine and describe a closed subvariety which contains all non-identifiable sextics. The description of the subvariety is geometric, but one can derive from that an algorithm which can guarantee that a given form is identifiable.

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arXiv2021

On the description of identifiable quartics

In this paper we study the identifiability of specific forms (symmetric tensors), with the target of extending recent methods for the case of $3$ variables to more general cases. In particular, we focus on forms of degree $4$ in $5$ variables. By means of tools coming from classical algebraic geometry, such as Hilbert function, liaison procedure and Serre's construction, we give a complete geometric description and criteria of identifiability for ranks $\geq 9$, filling the gap between rank $\leq 8$, covered by Kruskal's criterion, and $15$, the rank of a general quartic in $5$ variables. For the case $r=12$, we construct an effective algorithm that guarantees that a given decomposition is unique.

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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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