Łucja Farnik | University of the National Education Commission in Krakow (original) (raw)

Papers by Łucja Farnik

$Research paper thumbnail of Geproci sets on skew lines in <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">P</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb P^3</annotation></semantics></math>P3 with two transversals$

arXiv (Cornell University), Dec 6, 2023

$Research paper thumbnail of Geproci sets and the combinatorics of skew lines in <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">P</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb P^3</annotation></semantics></math>P3$

arXiv (Cornell University), Jul 31, 2023

arXiv (Cornell University), Oct 17, 2017

In this note we consider two configurations of twelve lines with nineteen triple points (i.e., po... more 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) ⊂ 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) 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.

arXiv (Cornell University), Mar 28, 2023

In this short note we develop new methods toward the ultimate goal of classifying geproci sets in... more In this short note we develop new methods toward the ultimate goal of classifying geproci sets in P 3. We apply these method 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 setup ; this is the context of follow-up work in progress. 1 Introduction The study of geproci sets was initialized in one of the previous workshops on Lefschetz properties held in Levico Terme (Italy) in 2018. In the present note we work exclusively over the field C of complex numbers. Definition 1.1 (A geproci set of points). We say that a set of points Z ⊂ P N C with N ≥ 3 is geproci (for GEneral PROjection is a Complete Intersection), if its general projection to a hyperplane is a complete intersection. We say that a geproci set is trivial, if it is already contained in a hyperplane. From now on we consider only nontrivial geproci sets. So far nontrivial geproci sets have been discovered only in P 3. They project to a plane, where their images are the intersection points of two curves of degrees a and b (equivalently: the ideal of the projection has exactly two generators: one of degree a and another of degree b). We assume that a ≤ b and we refer to such sets of points as (a, b)-geproci. Example 1.2 (A grid). Assume that we have two positive integers a ≤ b. Let Z ⊂ P 3 be a grid, i.e., the set of all intersection points among lines in two sets L = {L 1 ,. .. , L a } and M = {M 1 ,. .. , M b } such that lines from the same set, either L or M, are pairwise skew but any two lines from distinct sets intersect in a point. It is elementary to see that a grid is an (a, b)-geproci set for all values of a, b and the set is nontrivial for a, b ≥ 2 and it is contained in a unique quadric for a, b ≥ 3. Grids exist for any values of a and b. They were studied extensively in [2]. Here we are interested in geproci sets which are not grids but half grids. Definition 1.3 (A half grid). We say that a nontrivial (a, b)-geproci set Z ⊂ P 3 is a half grid, if it is not a grid but one of the curves determining its general projection as a complete intersection can be taken as a union of lines.

arXiv (Cornell University), Jul 10, 2019

The aim of this note is to give a generalization of some results concerning unexpected hypersurfa... more 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 P n × P n defined by determinants. We apply our results to unexpected hypersurfaces in 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.

Cornell University - arXiv, Sep 11, 2022

Introduction ix 0.1. General Context ix 0.2. History x 0.3. Questions and Results xi 0.4. Summary... more Introduction ix 0.1. General Context ix 0.2. History x 0.3. Questions and Results xi 0.4. Summary xii Chapter 1. Preliminaries 1 1.1. Harmonic points 1 1.2. Segre Embeddings and grids 5 1.3. Intersection of quadric cones in P 3 1.4. Arrangements of lines 1.5. Basic algebraic and geometric definitions used in this book Chapter 2. Weddle and Weddle-like varieties 2.1. The d-Weddle locus for a finite set of points in projective space 2.2. The d-Weddle scheme and two approaches to finding it 2.3. Different sets of six points and their Weddle schemes and loci 2.4. d-Weddle loci for some general sets of points in P 3 2.5. Connections to Lefschetz Properties Chapter 3. Geometry of the D 4 configuration 3.1. The rise of D 4 3.2. The geprociness of D 4 Chapter 4. The geography of geproci sets: a complete numerical classification 4.1. A standard construction 4.2. Classification of nondegenerate (a, b)-geproci sets for a ≤ 3 Chapter 5. Nonstandard geproci sets of points in projective space 5.1. Beyond the standard construction 5.2. The Klein configuration 5.3. The Penrose configuration 5.4. The H 4 configuration Chapter 6. Equivalences of geproci sets 6.1. Realizability over the real numbers 6.2. Projective and combinatorial equivalence of geproci sets Chapter 7. Unexpected cones 7.1. Geproci sets and unexpected cones 7.2. Unexpected cubic cones for B n+1 configurations of points 7.3. Unexpected cones for simplicial skeleta Chapter

The purpose of this note is to study configurations of lines in projective planes over arbitrary ... more The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields F over which such extremal configurations exist. We show that there does not exist a field admitting a configuration of 11 lines with 17 triple points, even though such a configuration is allowed combinatorially. Finally, we present an infinite series of configurations which have a high number of triple intersection points.

We study k-very ampleness of line bundles on blow-ups of hyperelliptic surfaces at r very general... more We study k-very ampleness of line bundles on blow-ups of hyperelliptic surfaces at r very general points. We obtain a numerical condition on the number of points for which a line bundle on the blow-up of a hyperelliptic surface at these r points gives an embedding of order k.

We study Seshadri constants of ample line bundles on hyperelliptic surfaces. We obtain new lower ... more We study Seshadri constants of ample line bundles on hyperelliptic surfaces. We obtain new lower bounds and compute the exact values of Seshadri constants in some cases. Our approach uses results of F. Serrano (1990), B. Harboune and J. Roe (2008), F. Bastianelli (2009), A.L. Knutsen, W. Syzdek and T. Szemberg (2009).

In this note we contribute to the study of Seshadri constants on abelian and bielliptic surfaces.... more In this note we contribute to the study of Seshadri constants on abelian and bielliptic surfaces. We specifically focus on bounds that hold on all such surfaces, depending only on the self-intersection of the ample line bundle under consideration. Our result improves previous bounds and it provides rational numbers as bounds, which are potential Seshadri constants.

In the paper we give an upper bound for the Waldschmidt constants of the wide class of ideals. Th... more In the paper we give an upper bound for the Waldschmidt constants of the wide class of ideals. This generalizes the result obtained by Dumnicki, Harbourne, Szemberg and Tutaj-Gasinska, Adv. Math. 2014. Our bound is given by a root of a suitable derivative of a certain polynomial associated with the asymptotic Hilbert polynomial.

The computation of the dimension of linear systems of plane curves through a bunch of given multi... more The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in Algebraic Geometry. In general, it is still an open problem to understand when the points fail to impose independent conditions. Despite many partial results, a complete solution is not known, even if the fixed points are in general position. The answer in the case of general points in the projective plane is predicted by the famous Segre-Harbourne-Gimigliano-Hirschowitz conjecture. When we consider fixed points in special position, even more interesting situations may occur. Recently Di Gennaro, Ilardi and Vallès discovered a special configuration Z of nine points with a remarkable property: a general triple point always fails to impose independent conditions on the ideal of Z in degree four. The peculiar structure and properties of this kind of unexpected curves were studied by Cook II, Harbourne, Migliore and Nagel. By using both explici...

$Research paper thumbnail of On the non-existence of orthogonal instanton bundles on P^{2n+1}$

Le Matematiche, 2009

In this paper we prove that there do not exist orthogonal instanton bundles on P ^{2n+1} . In ord... more In this paper we prove that there do not exist orthogonal instanton bundles on P ^{2n+1} . In order to demonstrate this fact, we propose a new way of representing the invariant, introduced by L. Costa and G. Ottaviani, related to a rank 2n instanton bundle on P ^{2n+1}.

Combinatorial Structures in Algebra and Geometry

Veneroni maps are a class of birational transformations of projective spaces. This class contains... more Veneroni maps are a class of birational transformations of projective spaces. This class contains the classical Cremona transformation of the plane, the cubo-cubic transformation of the space and the quatro-quartic transformation of P 4. Their common feature is that they are determined by linear systems of forms of degree n vanishing along n + 1 general flats of codimension 2 in P n. They have appeared recently in a work devoted to the so called unexpected hypersurfaces. The purpose of this work is to refresh the collective memory of the mathematical community about these somewhat forgotten transformations and to provide an elementary description of their basic properties given from a modern point of view.

$Research paper thumbnail of Rationality of Seshadri constants on general blow ups of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">P</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{P}^2</annotation></semantics></math>P2$

arXiv: Algebraic Geometry, 2019

Let XXX be a projective surface and let LLL be an ample line bundle on XXX. The global Seshadri c... more Let XXX be a projective surface and let LLL be an ample line bundle on XXX. The global Seshadri constant varepsilon(L)\varepsilon(L)varepsilon(L) of LLL is defined as the infimum of Seshadri constants varepsilon(L,x)\varepsilon(L,x)varepsilon(L,x) as xinXx\in XxinX varies. It is an interesting question to ask if varepsilon(L)\varepsilon(L)varepsilon(L) is a rational number for any pair (X,L)(X, L)(X,L). We study this question when XXX is a blow up of mathbbP2\mathbb{P}^2mathbbP2 at rge0r \ge 0rge0 very general points and LLL is an ample line bundle on XXX. For each rrr we define a textitsubmaximalitythreshold\textit{submaximality threshold}textitsubmaximalitythreshold which governs the rationality or irrationality of varepsilon(L)\varepsilon(L)varepsilon(L). We state a conjecture which strengthens the SHGH Conjecture and assuming that this conjecture is true we determine the submaximality threshold.

B\"or\"oczky configurations of lines have been recently considered in connection with t... more B\"or\"oczky 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 B\"or\"oczky configurations of 13, 14, 16, 18 and 24 lines and investigate rational points of these parameter spaces.

Starting with the pioneering work of Ein and Lazarsfeld [EinLaz93] restrictions on values of Sesh... more Starting with the pioneering work of Ein and Lazarsfeld [EinLaz93] restrictions on values of Seshadri constants on algebraic surfaces have been studied by many authors [Bau99, BauSze11, HarRoe08, KSS09, Nak05, Ste98, Sze12, Xu95]. In the present note we show how approximation involving continued fractions combined with recent results of Küronya and Lozovanu on Okounkov bodies of line bundles on surfaces [KurLoz14, KurLoz15] lead to effective statements considerably restricting possible values of Seshadri constants. These results in turn provide strong additional evidence to a conjecture governing the Seshadri constants on algebraic surfaces with Picard number 1.

Instanton bundles were introduced in [4] on P3 and in [7] on P2n+1. Recently many authors have be... more Instanton bundles were introduced in [4] on P3 and in [7] on P2n+1. Recently many authors have been dealing with them. The geometry of special instanton bundles is discussed in [8]. Authors of [3] study the stability of instanton bundles. They prove that all special symplectic instanton bundles on P2n+1 are stable and all instanton bundles on P5 are stable. Some computations involving instanton bundles can be done using computer programs, e.g. Macaulay, as it is presented in [2]. The SL(2)-action on the moduli space of special instanton bundles on P3 is described in [5]. In [6] it is shown that the moduli space of symplectic instanton bundles on P2n+1 is affine, by introducing the invariant that we use in this paper.

Journal of Algebraic Combinatorics

In this note, we consider two configurations of twelve lines with nineteen triple points (i.e. po... more 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 arrangemental combinatorial features, which means that 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) ⊂ I 2 holds, while for the other it does not. Hence, for ideals of points defined by arrangements of lines, the (non)containment of a symbolic power in an ordinary power is not determined alone by arrangemental combinatorial features of the configuration. Moreover, for the configuration with the non-containment I (3) 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.