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22 pages, 405 KB

Open AccessArticle

Multiple Points with Increasing Multiplicities on a Fixed Projective Set

by Edoardo Ballico

Symmetry 2026, 18(5), 877; https://doi.org/10.3390/sym18050877 - 21 May 2026

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Take a finite subset S of an n-dimensional projective space. We study the Hilbert function of the multiples

m S

of S, mainly when S is general or at least very general. We recall several classical conjectures on this problem, raise new [...] Read more.

Take a finite subset S of an n-dimensional projective space. We study the Hilbert function of the multiples

m S

of S, mainly when S is general or at least very general. We recall several classical conjectures on this problem, raise new open questions, and prove some particular cases. An open question is if all

m S

have the expected Hilbert function. We find cases in which there are Zariski open subsets of sets S with maximal rank for all m and pairs

(n, # S)

for which no such open set exists. We start the study of the m-Terracini sets proving when the first one is nonempty for Veronese embeddings. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2026)

9 pages, 270 KB

Open AccessArticle

Moduli Spaces of Arrangements of 12 Projective Lines with a Sextic Point

by Meirav Amram, Eran Lieberman, Sheng-Li Tan, Mina Teicher and Xiao-Hang Wu

Mathematics 2026, 14(6), 1052; https://doi.org/10.3390/math14061052 - 20 Mar 2026

Viewed by 443

Abstract

In this paper, we classify moduli spaces of arrangements of 12 lines with a sextic point. We show that these moduli spaces can consist of more than two connected components. We also present defining equations of arrangements whose moduli spaces are not irreducible, [...] Read more.

(This article belongs to the Special Issue Computational Geometry: Theory, Algorithms and Applications)

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9 pages, 286 KB

Open AccessArticle

Ranks with Respect to a Projective Variety and a Cost-Function

by Edoardo Ballico

AppliedMath 2022, 2(3), 457-465; https://doi.org/10.3390/appliedmath2030026 - 2 Aug 2022

Viewed by 1950

Abstract

Let

X \subset P^{r}

be an integral and non-degenerate variety. A “cost-function” (for the Zariski topology, the semialgebraic one, or the Euclidean one) is a semicontinuous function

w : = [1, + \infty) \cup + \infty

such that [...] Read more.

Let

X \subset P^{r}

be an integral and non-degenerate variety. A “cost-function” (for the Zariski topology, the semialgebraic one, or the Euclidean one) is a semicontinuous function

w : = [1, + \infty) \cup + \infty

such that

w (a) = 1

for a non-empty open subset of X. For any

q \in P^{r}

, the rank

r_{X, w} (q)

of q with respect to

(X, w)

is the minimum of all

\sum_{a \in S} w (a)

, where S is a finite subset of X spanning q. We have

r_{X, w} (q) < + \infty

for all q. We discuss this definition and classify extremal cases of pairs

(X, q)

. We give upper bounds for all

r_{X, w} (q)

(twice the generic rank) not depending on w. This notion is the generalization of the case in which the cost-function w is the constant function 1. In this case, the rank is a well-studied notion that covers the tensor rank of tensors of arbitrary formats (PARAFAC or CP decomposition) and the additive decomposition of forms. We also adapt to cost-functions the rank 1 decomposition of real tensors in which we allow pairs of complex conjugate rank 1 tensors. Full article

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