arXiv subjects

Tomasz Szemberg

Publications and source records attributed to Tomasz Szemberg.

At least 19 recordsLinked to original sources

arXiv2026

Demailly's Inequality for Finite Sets of Points in Positive Characteristic

Let $k$ be an algebraically closed field of characteristic $p>0$, $n\ge1$, and let $X\subset \mathbb P^n_k$ be a finite nonempty set of distinct points with the defining ideal $I=I(X)$. We give a proof of Demailly's inequality in positive characteristic. The argument is based on the Frobenius--Hasse derivative method used in this context by Hà and Sivakumar. The key additional observation is a strict-growth lemma for the initial degrees of symbolic powers of a finite set of affine points: $$ α(J^{(t)})\ge α(J^{(t-1)})+1\qquad(t\ge1). $$ In characteristic $p$, if a minimum-degree polynomial has a nonzero first ordinary derivative, this follows by differentiation; if all first ordinary derivatives vanish, perfectness gives a $p$th root and an induction on the symbolic exponent. The remainder of the proof uses the $q=p^e$ Frobenius decomposition and a maximal Hasse derivative.

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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

A Pascal-type construction of the Segre cubic and the Cremona--Richmond configuration

We present a Pascal-type residual construction in P^4. Starting from two quadruples of hyperplanes whose four diagonal intersection planes lie in a hyperplane, we show that the twelve residual planes lie on a cubic threefold. In the general case this cubic is the Segre cubic, and the construction recovers its fifteen planes and the associated Cremona--Richmond configuration. We also exhibit a point-line realization of this configuration in P^4 and show that it gives a (5,3)-geprofi set.

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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

Waldschmidt constants of symmetric sets of points in $\mathbb{P}^3$

Configurations of points defined by complex reflection groups have attracted a lot of attention recently in several directions of research, e.g., the containment problem between ordinary and symbolic powers of ideals, in the theory of unexpected hypersurfaces, in the study of sets of points whose general projections are complete intersections and in the ideas revolving around the Bounded Negativity Conjecture. In order to understand these configurations better several attempts have been undertaken to compute their various invariants. In this paper we focus on their Waldschmidt constants. In the case of plane configurations of points determined by reflection groups most (but not all) Waldschmidt constants are known. Here we pass to configurations in $\mathbb{P}^3$ where new ideas are required as the identification between divisors and curves is no longer available. In particular, we precisely compute the Waldschmidt constant for configurations of points in $\mathbb{P}^3$ coming from the $D_4,B_4,F_4$, and $H_4$ root systems.

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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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arXiv2024

Line arrangements with many triple points

In this paper, we construct an infinite series of line arrangements in characteristic two, each featuring only triple intersection points. This finding challenges the existing conjecture that suggests the existence of only a finite number of such arrangements, regardless of the characteristic. Leveraging the theory of matroids and employing computer algebra software, we rigorously examine the existence and non-existence across various characteristics of line arrangements with up to 19 lines maximizing the number of triple intersection points.

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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

On the number of vertices of Newton--Okounkov polygons

The Newton--Okounkov body of a big divisor D on a smooth surface is a numerical invariant in the form of a convex polygon. We study the geometric significance of the shape of Newton--Okounkov polygons of ample divisors, showing that they share several important properties of Newton polygons on toric surfaces. In concrete terms, sides of the polygon are associated to some particular irreducible curves, and their lengths are determined by the intersection numbers of these curves with D. As a consequence of our description we determine the numbers k such that D admits some k-gon as a Newton--Okounkov body, elucidating the relationship of these numbers with the Picard number of the surface, which was first hinted at by work of Küronya, Lozovanu and Maclean.

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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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