Hunting for Miyaoka-Kobayashi curves
We study the possible singularities of Miyaoka-Kobayashi curves and prove several results about the non-existence of such curves in degrees $\geq 8$.
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arXiv subjects
Piotr Pokora
Publications and source records attributed to Piotr Pokora.
Addition-deletion results for plus-one generated curves
In the recent paper A. Dimca proves that when one adds to or deletes a line from a free curve the resulting curve is either free or plus-one generated. We prove the converse statements, we give an additional insight into the original deletion result, and derive a characterisation of free curves in terms of behaviour to addition/deletion of lines. We describe the possible splitting types of the bundle of logarithmic vector fields associated to a plus-one generated curve.
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On free and nearly free arrangements of conics admitting certain ${\rm ADE}$ singularities
The main purpose of this paper is to provide combinatorial constraints on the constructability of free and nearly free arrangements of smooth plane conics admitting certain ${\rm ADE}$ singularites.
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Construction of free curves by adding lines to a given curve
In the present note we construct new families of free plane curves starting from a curve $C$ and adding high order inflectional tangent lines of $C$, lines joining the singularities of the curve $C$, or lines in the tangent cone of some singularities of $C$. These lines $L$ have in common that the intersection $C \cap L$ consists of a small number of points. We introduce the notion of a supersolvable plane curve and conjecture that such curves are always free, as in the known case of line arrangements. Some evidence for this conjecture is given as well, both in terms of a general result in the case of quasi homogeneous singularities and in terms of specific examples. We construct a new example of maximizing curve in degree 8 and the first and unique known example of maximizing curve in degree 9. In the final section, we use a stronger version of a result due to Schenck, Terao and Yoshinaga to construct families of free conic-line arrangements by adding lines to the conic-line arrangements of maximal Tjurina number recently classified by V. Beorchia and R. M. Miró-Roig in arXiv:2303.04665.
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On K-stability of $\mathbb{P}^3$ blown up along a quintic elliptic curve
In this note, we study K-stability of smooth Fano threefolds that can be obtained by blowing up the three-dimensional projective space along a smooth elliptic curve of degree five.
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Klein's arrangements of lines and conics
In this paper we construct several arrangements of lines and/or conics that are derived from the geometry of the Klein arrangement of $21$ lines in the complex projective plane.
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Maximizing curves viewed as free curves
The aim of this paper is to provide a direct link between maximizing curves that occur in the construction of smooth algebraic surfaces having the maximal possible Picard numbers and reduced free plane curves with simple singularities. We also investigate odd degree plane curves with simple singularities having maximal total Tjurina number.
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$\mathcal{Q}$-conic arrangements in the complex projective plane
We study the geometry of $\mathcal{Q}$-conic arrangements in the complex projective plane. These are arrangements consisting of smooth conics and they admit certain quasi-homogeneous singularities. We show that such $\mathcal{Q}$-conic arrangements are never free. Moreover, we provide combinatorial constraints of the weak combinatorics of such arrangements.
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Algebraic properties of Levi graphs associated with curve arrangements
In the present paper we study algebraic properties of edge ideals associated with plane curve arrangements via their Levi graphs. Using combinatorial properties of such Levi graphs we are able to describe those monomial algebras being Cohen-Macaulay, Buchsbaum, and sequentially Cohen-Macaulay. We also condsider the projective dimension and the Castelnuovo-Mumford regularity for these edge ideals. We provide effective lower and upper bounds on them. As a byproduct of our study we connect, in general, various Buchsbaum properties of squarefree modules.
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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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Special triple covers of algebraic surfaces
We study special triple covers $f\colon T \to S$ of algebraic surfaces, where the Tschirnhausen bundle $\mathcal E = \left(f_*\mathcal O_T/\mathcal O_S\right)^\vee$ is a quotient of a split rank three vector bundle, and we provide several necessary and sufficient criteria for the existence. As an application, we give a complete classification of special triple planes, finding among others two nice families of K3 surfaces.
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Conic-line arrangements in the complex projective plane
The main goal of this note is to begin a systematic study on conic-line arrangements in the complex projective plane. We show a de Bruijn-Erdős-type inequality and Hirzebruch-type inequality for a certain class of conic-line arrangements having ordinary singularities. We will also study, in detail, certain conic-line arrangements in the context of the geography of log-surfaces and free divisors in the sense of Saito.
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On plane conic arrangements with nodes and tacnodes
In the present paper, we study arrangements of smooth plane conics having only nodes and tacnodes as the singularities. We provide an interesting estimation on the number of nodes and tacnodes that depends only on a linear function of the number of conics. Based on that result, we obtain a new upper bound on the number of tacnodes which turns out to be better than Miyaoka's bound for a large enough number of conics. We also study the freeness and nearly freeness of such arrangements providing a detailed description.
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On submaximal curves in fake projective planes
The main purpose of the note is to exclude the existence of certain submaximal curves in fake projective planes. This will lead to lower bounds on multipoint Seshadri constants of `fake' $\mathcal{O}(1)$ on fake projective planes.
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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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Hirzebruch-type inequalities viewed as tools in combinatorics
The main purpose of this survey is to provide an introduction, algebro-topological in nature, to Hirzebuch-type inequalities for plane curve arrangements in the complex projective plane. These inequalities gain more and more interest due to their utility in many combinatorial problems related to point or line arrangements in the plane. We would like to present a summary of the technicalities and also some recent applications, for instance in the context of the Weak Dirac Conjecture. We also advertise some open problems and questions.
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On Seshadri constants and point-curve configurations
In the note we study the multipoint Seshadri constants of $\mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1)$ centered at singular loci of certain curve arrangements in the complex projective plane. Our first aim is to show that the values of Seshadri constants can be approximated with use of a combinatorial invariant which we call the configurational Seshadri constant. We study specific examples of point-curve configurations for which we provide actual values of the associated Seshadri constants. In particular, we provide an example based on Hesse point-conic configuration for which the associated Seshadri constant is computed by a line. This shows that multipoint Seshadri constants are not purely combinatorial.
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Towards the weighted bounded negativity conjecture for blow-ups of algebraic surfaces
In the present paper, we focus on a weighted version of the Bounded Negativity Conjecture which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a global constant. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.
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