arXiv subjects

Grzegorz Malara

Publications and source records attributed to Grzegorz Malara.

At least 19 recordsLinked to original sources

arXiv2026

From Demailly's Inequality to Harbourne--Huneke Containments for General Points

Let $I$ be the defining ideal of a general set of $s$ points in projective $N$-space over an algebraically closed field. Harbourne and Huneke conjectured the strengthened symbolic containment $$ I^{(Nr)}\subseteq \mathfrak{m}^{r(N-1)}I^r $$ for every $r\ge 1$. For general points this was known in low dimension and for small numbers of points, while in arbitrary dimension and arbitrary cardinality the best general result was the stable version, valid on one dense Zariski-open set for all sufficiently large $r$. We show that the recent proof of Demailly's conjecture for arbitrary finite point sets by Hà and Sivakumar removes the stability threshold. The key observation is that the strict generic Waldschmidt estimate of Bisui and Nguyen allows one to select a single finite symbolic level $m_0$. Equality of the corresponding interpolation number is a Zariski-open condition. Demailly's inequality then propagates this one finite condition to lower bounds for every symbolic power, and a standard regularity criterion converts those bounds into the Harbourne--Huneke containment for every $r$. Consequently, for every $N\ge2$ and every $s\ge1$, one dense Zariski-open family of $s$-point configurations in $\mathbb{P}^N$ satisfies the Harbourne--Huneke containment simultaneously for all $r\ge 1$. The argument also clarifies a quantifier issue in the earlier literature: before the full Demailly theorem, estimates on the generic fibre naturally yielded either all $r$ on a very general set, or one open set only for $r\gg 0$. The new theorem replaces infinitely many symbolic conditions by a single finite interpolation condition.

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arXiv2026

Dihedral reflections and an infinite series of irrational Seshadri constants

Laface and Ugaglia recently constructed an irrational one-point Seshadri constant on the blow-up of $\mathbb{P}^2$ at nine very general points by combining a dihedral orbit on $\mathbb{P}^1\times\mathbb{P}^1$, a sequence of de Jonquières transformations, and a reflection argument along a $(-4)$-curve with balanced normal bundle. We show that the same mechanism extends uniformly to every odd integer $n\geq 5$. For $n=2k+1$ we prove $$ \varepsilon\bigl(\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(n-4,1);p_1,\ldots,p_{2n}\bigr)=\sqrt{\frac{n-4}{n}} $$ at $2n$ very general points, and already at a very general free orbit of a fixed dihedral group of order $2n$. For every odd $n\geq 7$ this produces an explicit ample line bundle on the blow-up of $\mathbb{P}^2$ at $k+7=(n+13)/2$ very general points whose one-point Seshadri constant at a very general point equals $$ 2\sqrt{n(n-4)}. $$ More precisely, after one quadratic transformation we obtain the ample divisor $$ L_n=(3n-4)H-nE_1-(n-2)(E_2+\cdots+E_5)-4(E_6+\cdots+E_{k+5})-2(E_{k+6}+E_{k+7}), $$ with $L_n^2=4n(n-4)$ and $\varepsilon(L_n;x)=\sqrt{L_n^2}$. We also isolate an abstract balanced-reflection principle underlying the construction: a nef class on a special fiber can be reflected across a rational curve of square $-2a$ whenever the curve has normal bundle $\mathcal{O}_{\mathbb{P}^1}(-a)^{\oplus 2}$ in the total space, and the reflected class is nef on very general fibers.

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arXiv2026

On the Geometry of Sixers on the Fermat Cubic Surface

A sixer is a configuration of six pairwise skew lines on a smooth cubic surface, equivalently a choice of six exceptional curves defining a blow-down to $\mathbb{P}^2$. We study the $72$ sixers on the Fermat cubic surface $x_0^3+x_1^3+x_2^3+x_3^3=0$. We show that these sixers split into two orbits under the automorphism group of the Fermat cubic, of sizes $18$ and $54$. We give a geometric interpretation of this decomposition through the corresponding plane blow-up models: representatives of the two orbit types determine six-point configurations in $\mathbb{P}^2$ whose projective automorphism groups have orders $36$ and $12$, respectively. These groups identify with the stabilizers of the corresponding sixers and recover the two orbit sizes. We then compute the projective groups associated with representatives of the two orbits over $K=\mathbb Q(ω)$, where $ω^2+ω+1=0$, and distinguish them arithmetically by the determinant square-class character $δ_K:\mathrm{PGL}_2(K)\longrightarrow K^*/(K^*)^2$. Its images have $\mathbb F_2$-dimensions $1$ and $2$ for the orbits of sizes $18$ and $54$, respectively. Modulo $13$, the corresponding finite images are $\mathrm{PSL}_2(\mathbb F_{13})$ and $\mathrm{PGL}_2(\mathbb F_{13})$, respectively, and the determinant-character distinction persists for all choices of normalization triple.

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arXiv2026

Interpolation matrices and jumping lines of logarithmic bundles

We study jumping lines loci of logarithmic bundles associated with finite sets of points in the projective plane. Using the interpolation matrix introduced in [DMTG25], we describe these loci as the zero sets of explicit determinants depending on parameters $(d,m)$ determined by the number of points. We show that for points in general position the determinant defines an irreducible curve of the expected degree, while for special configurations it acquires fixed components related to the combinatorics of the arrangement. The approach provides a new geometric interpretation of the classical jumping lines of Dolgachev--Kapranov and Barth, and connects them to the framework of unexpected curves and hypersurfaces.

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arXiv2025

Unexpected hypersurfaces of type $(d+k,d)$

Unexpected hypersurfaces arise when vanishing in points of a set $Z$ and higher-order vanishing along a general linear subspace fails to impose the expected number of independent conditions on forms of a fixed degree. The phenomenon was first observed for planar curves by Cook, Harbourne, Migliore and Nagel. This paper shows a syzygy-based construction of, possibly unexpected, hypersurfaces of degree $d+k$ in $\mathbb{P}^n$, vanishing along a codimension two general linear subspace with multiplicity $d$; thus generalizing the work of Trok and the previous work of the last two authors. Our framework unifies the classical planar cases with higher-dimensional examples, including Trok's construction. We give a sufficient criterion for unexpectedness (via the splitting behaviour the syzygy bundles of the powers of the Jacobian ideal, associated with the hyperplane arrangement dual to $Z$) and provide explicit examples in $\mathbb{P}^3$ and $\mathbb{P}^4$.

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arXiv2025

Matrixwise (approach to unexpected hypersurfaces) Reloaded

In the paper we provide a new method of proving the existence of a hypersurface of degree $d$ in $\mathbb{P}^n$, with a general point of multiplicity $m$ and vanishing at a given set of points $Z$, by looking at weak combinatorics of a set $Z$. This method has a direct application in the theory of unexpected hypersurfaces, where many of the examples are based only on computer experiments.

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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

Construction of free curves by adding osculating conics to a given cubic curve

In the present article we construct new families of free and nearly free curves starting from a plane cubic curve $C$ and adding some of its hyperosculating conics. We present results that involve nodal cubic curves and the Fermat cubic. In addition, we provide new insight into the geometry of the $27$ hyperosculating conics of the Fermat cubic curve using well-chosen group actions.

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arXiv2022

On unexpected curves of type $(d+k,d)$

We present a construction explaining the existence of (unexpected) curves of degree $d+k$, passing through a set $Z$ of points on $\mathbb{P}^2$, and having a generic point $P$ of multiplicity $d$. The construction is based on the syzygies of the $k$-th powers of Jacobian of the product of lines dual to the points of $Z$. We prove also a result characterizing the unexpectedness of the curves via splitting type of the bundle of these syzygies retricted to the line dual to $P$.

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arXiv2021

On $3$-syzygy and unexpected plane curves

In this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study $3$-syzygy arrangements and we present examples that admit unexpected curves.

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arXiv2020

Regularity and symbolic defect of points on rational normal curves

In this paper we study ideals of points lying on rational normal curves defined in projective plane and projective $ 3 $-space. We give an explicit formula for the value of Castelnuovo-Mumford regularity for their ordinary powers. Moreover, we compare the $ m $-th symbolic and ordinary powers for such ideals in order to show whenever the $ m $-th symbolic defect is non-zero.

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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

Asymptotic invariants of generic initial ideals

Generic initial ideals (gins in short) were systematically introduced by Galligo in 1974 under the name of Grauert invariants since they appeared apparently first in works of Grauert and Hironaka. Ever since they are of interest in commutative algebra and indirectly in algebraic geometry. Recently Mayes in a series of articles associated to gins geometric objects called limiting shapes. The construction resembles that of Okunkov bodies but there are some differences as well. This work is motivated by Mayes articles and explores the connections between gins, limiting shapes and some asymptotic invariants of homogeneous ideals, e.g. asymptotic regularity, Waldschmidt constant and some new invariants, which seem relevant from geometric point of view. In this note we generalize Mayes ideas to graded families of ideals. We work out, sometimes surprising, properties of defined objects and invariants. For example we establish the existence of limiting shapes in dimension 2 which are polygons with an arbitrary high number of edges and vertices with irrational coordinates.

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arXiv2019

A matrixwise approach to unexpected hypersurfaces

The aim of this note is to give a generalization of some results concerning unexpected hypersurfaces. Unexpected hypersurfaces occur when the actual dimension of the space of forms satisfying certain vanishing data is positive and the imposed vanishing conditions are not independent. The first instance studied were unexpected curves in the paper by Cook II, Harbourne, Migliore, Nagel. Unexpected hypersurfaces were then investigated by Bauer, Malara, Szpond and Szemberg, followed by Harbourne, Migliore, Nagel and Teitler who introduced the notion of BMSS duality and showed it holds in some cases (such as certain plane curves and, in higher dimensions, for certain cones). They ask to what extent such a duality holds in general. In this paper, working over a field of characteristic zero, we study hypersurfaces in $\mathbb{P}^n\times\mathbb{P}^n$ defined by determinants. We apply our results to unexpected hypersurfacesin the case that the actual dimension is 1 (i.e., there is a unique unexpected hypersurface). In this case, we show that a version of BMSS duality always holds, as a consequence of fundamental properties of determinants.

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arXiv2018

New phenomena in the containment problem for simplicial arrangements

In this note we consider two simplicial arrangements of lines and ideals $I$ of intersection points of these lines. There are $127$ intersection points in both cases and the numbers $t_i$ of points lying on exactly $i$ configuration lines (points of multiplicity $i$) coincide. We show that in one of these examples the containment $I^{(3)} \subseteq I^2$ holds, whereas it fails in the other. We also show that the containment fails for a subarrrangement of $21$ lines. The interest in the containment relation between $I^{(3)}$ and $I^2$ for ideals of points in $¶^2$ is motivated by a question posted by Huneke around $2000$. Configurations of points with $I^{(3)} \not\subseteq I^2$ are quite rare. Our example reveals two particular features: All points are defined over $\Q$ and all intersection points of lines are involved. In examples studied by now only points with multiplicity $i\geq 3$ were considered. The novelty of our arrangements lies in the geometry of the element in $I^{(3)}$ which witness the noncontainment in $I^2$. In all previous examples such an element was a product of linear forms. Now, in both cases there is an irreducible curve of higher degree involved.

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arXiv2018

Quartic unexpected curves and surfaces

Our research is motivated by recent work of Cook II, Harbourne, Migliore, and Nagel on configurations of points in the projective plane with properties that are unexpected from the point of view of the postulation theory. In this note, we revisit the basic configuration of nine points appearing in work of Gennaro/Ilardi/Vallès and Harbourne, and we exhibit some additional new properties of this configuration. We then pass to projective three-space and exhibit a surface with unexpected postulation properties there. Such higher dimensional phenomena have not been observed so far.

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arXiv2018

Unexpected curves arising from special line arrangements

In a recent paper arXiv:1602.02300v2, Cook II, Harbourne, Migliore and Nagel related the splitting type of a line arrangement in the projective plane to the number of conditions imposed by a general fat point of multiplicity $j$ to the linear system of curves of degree $j+1$ passing through the configuration of points dual to the given arrangement. If the number of conditions is less than the expected, we say that the configuration of points admits unexpected curves. In this paper, we characterize supersolvable line arrangements whose dual configuration admits unexpected curves and we provide other infinite families of line arrangements with this property.

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arXiv2017

Fermat-type configurations of lines in $\mathbb P^3$ and the containment problem

The purpose of this note is to show a new series of examples of homogeneous ideals $I$ in ${\mathbb K}[x,y,z,w]$ for which the containment $I^{(3)}\subset I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb P}^3$, which resemble Fermat configurations of points in ${\mathbb P}^2$, see \cite{NagSec16}. All examples exhibiting the failure of the containment $I^{(3)}\subseteq I^2$ constructed so far have been supported on points or cones over configurations of points. Apart of providing new counterexamples, these ideals seem quite interesting on their own.

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