arXiv subjects

Justyna Szpond

Publications and source records attributed to Justyna Szpond.

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

From a Pascal construction to the Burkhardt quartic

We continue the study of Pascal-type residual constructions in projective four-space. Starting from two $k$-tuples of hyperplanes in $\mathbb P^4$ such that the $k$ diagonal intersection planes are contained in a hyperplane, one obtains a residual hypersurface of degree $k-1$ containing the remaining $k^2-k$ planes. In this work we consider the case $k=5$, where the twenty residual planes are contained in a quartic threefold. A balanced specialization of this construction is projectively equivalent to the celebrated Burkhardt quartic. In this model the twenty residual planes form one half of the forty Jacobi planes on the Burkhardt quartic. We reveal their incidence structure as governed by the directed complete graph on five vertices. The forty nodes naturally forced by these planes split as $30+10$, and the Burkhardt specialization adds five further nodes. We also write down the complementary twenty Jacobi planes explicitly and describe all forty Steiner hyperplanes in Pascal coordinates.

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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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arXiv2026

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 Miró-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

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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arXiv2024

Postulation of lines in P3 revisited

The purpose of the present note is to provide a new proof ot the well-known result due to Hartshorne and Hirschowitz to the effect that general lines in projective spaces have good postulation. Our approach uses specialization to a hyperplane and thus opens door to study postulation of general codimension 2 linear subspaces in projective spaces.

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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

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

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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arXiv2022

Sextactic points on the Fermat cubic curve and arrangements of conics

The purpose of this note is to report, in narrative rather than rigorous style, about the nice geometry of $6$-division points on the Fermat cubic $F$ and various conics naturally attached to them. Most facts presented here were derived by symbolic algebra programs and the idea of the note is to propose a research direction for searching for conceptual proofs of facts stated here and their generalisations. Extensions in several directions seem possible (taking curves of higher degree and contact to $F$, studying higher degree curves passing through higher order division points on $F$, studying curves passing through intersection points of already constructed curves, taking the duals etc.) and we hope some younger colleagues might find pleasure in following proposed paths as well as finding their own.

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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

Unexpected properties of the Klein configuration of $60$ points in ${\mathbb P}^3$

Felix Klein in course of his study of the regular icosahedron and its symmetries encountered a highly symmetric configuration of $60$ points in ${\mathbb P}^3$. This configuration has appeared in various guises, perhaps post notably as the configuration of points dual to the $60$ reflection planes in the group $G_{31}$ in the Shephard-Todd list. In the present note we show that the $60$ points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree $6$. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated to the configuration of $60$ points is a cone with a single singularity of multiplicity $6$ and the other has three singular points of multiplicities $4,2$ and $2$. Secondly, Chiantini and Migliore observed in 2020 that there are non-trivial sets of points in ${\mathbb P}^3$ with the surprising property that their general projection to ${\mathbb P}^2$ is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of $24$ points in ${\mathbb P}^3$ which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property. \

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arXiv2022

Companion varieties for root systems and Fermat arrangements

Unexpected hypersurfaces are a brand name for some special linear systems. They were introduced around 2017 and are a field of intensive study since then. They attracted a lot of attention because of their close tights to various other areas of mathematics including vector bundles, arrangements of hyperplanes, geometry of projective varieties. Our research is motivated by the what is now known as the BMSS duality, which is a new way of deriving projective varieties out of already constructed. The last author coined the concept of companion surfaces in the setting of unexpected curves admitted by the $B_3$ root system. Here we extend this construction in various directions. We revisit the configurations of points associated to either root systems or to Fermat arrangements and we study the geometry of the associated varieties and their companions.

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arXiv2020

Symbolic powers of codimension two Cohen-Macaulay ideals

Let $I_X$ be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme $X \subseteq \mathbb{P}^n$, and let $I_X^{(m)}$ denote its $m$-th symbolic power. We are interested in when $I_X^{(m)} = I_X^m$. We survey what is known about this problem when $X$ is locally a complete intersection, and in particular, we review the classification of when $I_X^{(m)} = I_X^m$ for all $m \geq 1$. We then discuss how one might weaken these hypotheses, but still obtain equality between the symbolic and ordinary powers. Finally, we show that this classification allows one to: (1) simplify known results about symbolic powers of ideals of points in $\mathbb{P}^1 \times \mathbb{P}^1$; (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of Römer.

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arXiv2020

Negative curves on special rational surfaces

We study negative curves on surfaces obtained by blowing up special configurations of points in the complex projective palne. Our main results concern the following configurations: very general points on a cubic, 3-torsion points on an elliptic curve and nine Fermat points. As a consequence of our analysis, we also show that the Bounded Negativity Conjecture holds for the surfaces we consider. The note contains also some problems for future attention.

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arXiv2019

Unexpected hypersurfaces with multiple fat points

Starting with the ground-breaking work of Cook II, Harbourne, Migliore and Nagel, there has been a lot of interest in unexpected hypersurfaces. In the last couple of months a considerable number of new examples and new phenomena has been observed and reported on. All examples studied so far had just one fat point. In this note we introduce a new series of examples, which establishes for the first time the existence of unexpected hypersurfaces with multiple fat points. The key underlying idea is to study Fermat-type configurations of points in projective spaces.

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arXiv2019

Fermat-type arrangements

The purpose of this work is to collect in one place available information on line arrangements known in the literature as braid, monomial, Ceva or Fermat arrangement. They have been studied for a long time and appeared recently in connection with highly interesting problems, namely: the containment problem between symbolic and ordinary powers of ideals and the existence of unexpected hypersurfaces. We also study also derived configurations of points (or more general: linear flats) which arise by intersecting hyperplanes in Fermat arrangements or by taking duals of these hyperplanes. Furthermore we discuss briefly higher dimensional generalizations and present results arising by applying this approach to problems mentioned above. Some of our results are original and appear for the first time in print.

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