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Halszka Tutaj-Gasińska

Publications and source records attributed to Halszka Tutaj-Gasińska.

13 recordsLinked to original sources
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

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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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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arXiv2019

New constructions of unexpected hypersurfaces in $\mathbb{P}^n$

In the paper we present new examples of unexpected varieties. The research on unexpected varieties started with a paper of Cook II, Harbourne, Migliore and Nagel and was continued in the paper of Harbourne, Migliore, Nagel and Teitler. Here we present three ways of producing unexpected varieties that expand on what was previously known. In the paper of Harbourne, Migliore, Nagel and Teitler, cones on varieties of codimension 2 were used to produce unexpected hypersurfaces. Here we show that cones on positive dimensional varieties of codimension 2 or more almost always give unexpected hypersurfaces. For non-cones, almost all previous work has been for unexpected hypersurfaces coming from finite sets of points. Here we construct unexpected surfaces coming from lines in $\mathbb{P}^3$, and we generalize the construction using birational transformations to obtain unexpected hypersurfaces in higher dimensions.

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arXiv2018

On the parameter space of Böröczky configurations

Böröczky configurations of lines have been recently considered in connection with the problem of the containment between symbolic and ordinary powers of ideals. Here we describe parameter families of Böröczky configurations of 13, 14, 16, 18 and 24 lines and investigate rational points of these parameter spaces.

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arXiv2017

Containment problem and combinatorics

In this note we consider two configurations of twelve lines with nineteen triple points (i.e., points where three lines meet). Both of them have the same combinatorial features. In both configurations nine of twelve lines have five triple points and one double point, and the remaining three lines have four triple points and three double points. Taking the ideal of the triple points of these configurations we discover that, quite surprisingly, for one of the configurations the containment $I^{(3)} \subset I^2$ holds, while for the other it does not. Hence for ideals of points defined by configurations of lines the (non)containment of a symbolic power in an ordinary power is not determined alone by combinatorial features of the arrangement. Moreover, for the configuration with the non-containment $I^{(3)} \nsubseteq I^2$ we examine its parameter space, which turns out to be a rational curve, and thus establish the existence of a rational non-containment configuration of points. Such rational examples are very rare.

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arXiv2015

Harbourne constants and conic configurations on the projective plane

In this note we exhibit the so-called Harbourne constants which capture and measure the Bounded Negativity on various birational models of an algebraic surface. We show an estimation for Harbourne constants for conic configurations on the complex projective plane with transversal intersection points.

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arXiv2014

Resurgences for ideals of special point configurations in ${\bf P}^N$ coming from hyperplane arrangements

Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence. There have been exciting new developments in this area recently. It had been expected for several years that $I^{Nr-N+1}\subseteq I^r$ should hold for the ideal $I$ of any finite set of points in ${\bf P}^N$ for all $r>0$, but in the last year various counterexamples have now been constructed, all involving point sets coming from hyperplane arrangements. In the present work, we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal.

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arXiv2013

Points fattening on P^1 x P^1 and symbolic powers of bi-homogeneous ideals

We study symbolic powers of bi-homogeneous ideals of points in the Cartesian product of two projective lines and extend to this setting results on the effect of points fattening obtained by Bocci, Chiantini and Dumnicki, Szemberg, Tutaj-Gasińska. We prove a Chudnovsky-type theorem for bi-homogeneous ideals and apply it to classification of configurations of points with minimal or no fattening effect. We hope that the ideas developed in this project will find further algebraic and geometric applications e.g. to study similar problems on arbitrary surfaces.

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arXiv2012

Linear subspaces, symbolic powers and Nagata type conjectures

Prompted by results of Guardo, Van Tuyl and the second author for lines in projective 3 space, we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r dimensional planes in projective n space for n at least 2r+1. These considerations lead to new conjectures that suggest that the well known conjecture of Nagata for points in the projective plane is not sporadic, but rather a special case of a more general phenomenon.

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