Searcharxiv⌕ Search

arXiv subjects

Justyna Szpond

Publications and source records attributed to Justyna Szpond.

At least 37 records · Page 2Linked to original sources
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.

math.AG↗
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.

math.AG↗
arXiv2019

Unexpected curves and Togliatti-type surfaces

The purpose of this note is to establish a direct link between the theory of unexpected hypersurfaces and varieties with defective osculating behavior. We identify unexpected plane curves of degree 4 as sections of a rational surface X of degree 7 in P^5 with its osculating spaces of order 2 which in every point of X have dimension lower than expected. We put this result in perspective with earlier examples of surfaces with defective osculating spaces due to Shifrin and Togliatti. Our considerations are rendered by an analysis of Lefschetz Properties of ideals associated with the studied surfaces.

math.AG↗
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.

math.AG↗
arXiv2018

Lower bounds for Waldschmidt constants of generic lines in $\mathbb{P}^3$ and a Chudnovsky-type theorem

The Waldschmidt constant $\alphahat(I)$ of a radical ideal $I$ in the coordinate ring of $\PP^N$ measures (asymptotically) the degree of a hypersurface passing through the set defined by $I$ in $\PP^N$. Nagata's approach to the 14th Hilbert Problem was based on computing such constant for the set of points in $\PP^2$. Since then, these constants drew much attention, but still there are no methods to compute them (except for trivial cases). Therefore the research focuses on looking for accurate bounds for $\alphahat(I)$. In the paper we deal with $\alphahat(s)$, the Waldschmidt constant for $s$ very general lines in $\PP^3$. We prove that $\alphahat(s) \geq \lfloor\sqrt{2s-1}\rfloor$ holds for all $s$, whereas the much stronger bound $\alphahat(s) \geq \lfloor\sqrt{2.5 s}\rfloor$ holds for all $s$ but $s=4$, $7$ and $10$. We also provide an algorithm which gives even better bounds for $\alphahat(s)$, very close to the known upper bounds, which are conjecturally equal to $\alphahat(s)$ for $s$ large enough.

math.AG↗
arXiv2018

Local effectivity in projective spaces

In this note we introduce a Waldschmidt decomposition of divisors which might be viewed as a generalization of Zariski decomposition based on the effectivity rather than the nefness of divisors. As an immediate application we prove a recursive formula providing new effective lower bounds on Waldschmidt constants of very general points in projective spaces. We use these bounds in order to verify Demailly's conjecture in a number of new cases.

math.AG↗
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.

math.AG↗
arXiv2017

On the postulation of lines and a fat line

In this note we show that the union of $r$ general lines and one fat line in ${\mathbb P}^3$ imposes independent conditions on forms of sufficiently high degree $d$, where the bound on $d$ is independent of the number of lines. This extends former results of Hartshorne and Hirschowitz on unions of general lines, and of Aladpoosh on unions of general lines and one double line.

math.AG↗
arXiv2017

On codimension two flats in Fermat-type arrangements

In the present note we study certain arrangements of codimension $2$ flats in projective spaces, we call them "Fermat arrangements". We describe algebraic properties of their defining ideals. In particular, we show that they provide counterexamples to an expected containment relation between ordinary and symbolic powers of homogeneous ideals.

math.AG↗
arXiv2017

Waldschmidt constants for Stanley-Reisner ideals of a class of graphs

In the present note we study Waldschmidt constants of Stanley-Reisner ideals of a hypergraph and a graph with vertices forming a bipyramid over a planar n-gon. The case of the hypergraph has been studied by Bocci and Franci. We reprove their main result. The case of the graph is new. Interestingly, both cases provide series of ideals with Waldschmidt constants descending to 1. It would be interesting to known if there are bounded ascending sequences of Waldschmidt constants.

math.AG↗
arXiv2016

Restrictions on Seshadri constants on surfaces

Starting with the pioneering work of Ein and Lazarsfeld restrictions on values of Seshadri constants on algebraic surfaces have been studied by many authors. In the present note we show how approximation involving continued fractions combined with recent results of Kuronya and Lozovanu on Okounkov bodies of line bundles on surfaces lead to effective statements considerably restricting possible values of Seshadri constants. These results in turn provide strong aditional evidence to a conjecture governing the Seshadri constants on algebraic surfaces with Picard number 1.

math.AG↗
arXiv2016

From Pappus Theorem to parameter spaces of some extremal line point configurations and applications

In the present work we study parameter spaces of two line point configurations introduced by Böröczky. These configurations are extremal from the point of view of Dirac-Motzkin Conjecture settled recently by Green and Tao. They have appeared also recently in commutative algebra in connection with the containment problem for symbolic and ordinary powers of homogeneous ideals and in algebraic geometry in considerations revolving around the Bounded Negativity Conjecture. Our main results are Theorem A and Theorem B. We show that the parameter space of what we call $B12$ configurations is a three dimensional rational variety. As a consequence we derive the existence of a three dimensional family of rational $B12$ configurations. On the other hand the moduli space of $B15$ configurations is shown to be an elliptic curve with only finitely many rational points, all corresponding to degenerate configurations. Thus, somewhat surprisingly, we conclude that there are no rational $B15$ configurations.

math.AG↗
arXiv2016

On absolute linear Harbourne constants

In the present note we study absolute linear Harbourne constants. These are invariants which were introduced in order to relate the lower bounds on the selfintersection of negative curves on birationally equivalent surfaces to the complexity of the birational map between them. We provide various lower and upper bounds on Harbourne constants and give their values for the number of lines $s$ of the form $p^{2r}+p^r+1$ for any prime number $p$ and also for all values of $s$ up to $31$. This extends considerably earlier results of the third author.

math.AG↗
arXiv2016

On the containment problem

The purpose of this note is to provide an overview of the containment problem for symbolic and ordinary powers of homogeneous ideals, related conjectures and examples. We focus here on ideals with zero dimensional support. This is an area of ongoing active research. We conclude the note with a list of potential promising paths of further research.

math.AG↗
arXiv2015

On linear Harbourne constants

In this note we compute values of global linear Harbourne constants over arbitrary fields for up to ten lines. These invariants have appeared recently in the discussions around the Bounded Negativity Conjecture. They seem to be of independent interest also from the point of view of combinatorics.

math.AG↗
arXiv2014

On the Sylvester-Gallai theorem for conics

In the present note we give a new proof of a result due to Wiseman and Wilson which establishes an analogue of the Sylvester-Gallai theorem valid for curves of degree two. The main ingredients of the proof come from algebraic geometry. Specifically, we use Cremona transformation of the projective plane and Hirzebruch inequality.

math.AG↗