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Piotr Pokora

Publications and source records attributed to Piotr Pokora.

At least 19 recordsLinked to original sources
arXiv2026

Simplicial arrangements in real projective three-space revisited

In this paper we study irreducible simplicial arrangements of projective planes in $\mathbb{P}^{3}(\mathbb{R})$ from combinatorial and projective geometry viewpoints. We first formulate a simpliciality criterion in terms of incidences between rank-two and rank-three flats, together with equivalent formulations using face numbers, reduced restrictions, and characteristic polynomials. We also relate the classical planar restriction data of Gr\"unbaum-Shephard to Ziegler multirestrictions. Our principal result concerns the rank-four special-vertex property: among the irreducible crystallographic Coxeter arrangements of rank four, the arrangements of types $A_4$ and $B_4$ admit a special vertex, whereas those of types $D_4$ and $F_4$ do not. A simplicial deletion chain inside $B_4$ supplies further irreducible examples with a special vertex. Finally, we compare rank-flat, Purdy-type, and Gr\"unbaum-Shephard defects for these arrangements.

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arXiv2026

On the Numerical Terao Conjecture

We prove that the Numerical Terao Conjecture holds for even-degree conic-line arrangements having only ADE singularities. We then show that the conjecture fails in the broader quasi-homogeneous setting once ordinary quadruple points are admitted, by constructing a degree-nine counterexample consisting of a pair of conic-line arrangements, each with seven lines and one smooth conic, with the same weak combinatorics \[ W(\mathcal{CL}) = (7,1;\,8A_1+D_4+4X_9). \] We show that one curve is free with exponents $(4,4)$, whereas the other is nearly free with exponents $(3,6)$. This yields a counterexample to the strongest known formulation of the Numerical Terao Conjecture.

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arXiv2026

A counterexample to Purdy's inequality for hyperplane arrangements in projective three-space

We record an explicit counterexample to a refined form of Purdy's inequality for essential hyperplane arrangements in projective three-space. Let $\mathcal{A}$ be an arrangement of $n$ hyperplanes in $\mathbb{P}^3_{\mathbb{C}}$. Let $\ell$ be the number of distinct intersection lines of $\mathcal{A}$, and let $p$ be the number of intersection points, where an intersection point means a point at which at least three hyperplanes meet. The expected inequality is \[ p-\ell+n+2\geq 0. \] The classical obstruction is the rank $2+2$ product arrangement, or dually a configuration of points contained in two skew lines. We explain this obstruction first, and then show that it is not the only one. The reflection-arrangement search leads naturally to a subarrangement of the monomial reflection arrangement of type $G(3,3,4)$. Looking dually, this configuration is not contained in two skew lines, and has \[ f_0(S)=12,\qquad f_1(S)=58,\qquad f_2(S)=43. \] Therefore its dual arrangement has \[ n=12,\qquad \ell=58,\qquad p=43, \] and hence \[ p-\ell+n+2=-1. \] Thus the refined statement excluding only the two-skew-lines obstruction is false.

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arXiv2026

A $(24_4,32_3)$-configuration on the Schur quartic with logarithmic Chern slope $14/5$

Let $X\subset\mathbb{P}^3$ be the Schur quartic \[ x_0^4-x_0x_1^3-x_2^4+x_2x_3^3=0. \] We exhibit a connected arrangement of $24$ lines on $X$, defined over $\mathbb{Q}(\sqrt{-3})$, whose singular locus consists of $32$ ordinary triple points and no other intersections. Each line contains four triple points. The resulting reduced divisor $D$ satisfies $D\sim6H$, where $H$ is the hyperplane class. If $\pi:Y\to X$ blows up the triple points and $B=(\pi^{-1}D)_{\mathrm{red}}$, then \[ \overline{c}_{1}^{2}(Y,B)=112,\qquad \overline{c}_{2}(Y,B)=40, \qquad \frac{\overline{c}_{1}^{2}(Y,B)}{\overline{c}_{2}(Y,B)}=\frac{14}{5}. \] This gives a negative answer to the K3-surface specialization of the proposed $8/3$ bound for transversal arrangements of rational curves. The configuration is one half of the $48$ lines of the second kind on $X$; an explicit projective automorphism exchanges the two halves. We deliver the line parametrizations and all $32$ triple-point coordinates. Ancillary exact-arithmetic data record the $120$ line-containment coefficients and all $276$ pair-incidence determinants. A finite-field mixed-integer search is described only as the discovery procedure and is not used in the proof.

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arXiv2026

A nine-line counterexample to a conjecture on the minimal degree of Jacobian relations

We construct two arrangements of nine lines in the complex projective plane with isomorphic intersection lattices but with different minimal degrees of Jacobian relations. The common weak combinatorics is \[ (n_2,n_3,n_4)=(9,7,1), \] so the example is not the classical Ziegler-Yuzvinsky pair, whose weak combinatorics is $(n_{2},n_{3}) = (18,6)$. For the two defining equations $f$ and $g$ we prove \[ {\rm mdr}(f)=4,\qquad {\rm mdr}(g)=5. \] Since the degree is $d=9$, the first equality gives ${\rm mdr}(f)<d/2$. Hence the pair gives a counterexample to the Generalized Terao Conjecture.

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arXiv2026

On Ziegler pairs of line arrangements: from non-existence to abundance

We study Ziegler pairs of line arrangements from both numerical and homological perspectives. We distinguish classical Ziegler pairs, for which the minimal degree of a Jacobian relation denoted by ${\rm mdr}$ differs, from the more general notion detected by distinct graded Betti data. First, we show that for arrangements of $d<9$ lines the intersection lattice determines ${\rm mdr}$, and hence classical Ziegler pairs do not occur in this range. Then we list several distinct Ziegler pairs with $d=10$ in the broader sense. In particular, we construct higher-degree examples with the same intersection lattice, the same minimal degree of a Jacobian relation, and the same Hilbert function of the Milnor algebra, but with different minimal graded free resolutions. Another rather surprising consequence is that being maximal Tjurina for a line arrangement is not determined by the combinatorics.

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arXiv2026

A Pascal-type construction of the Segre cubic and the Cremona--Richmond configuration

We present a Pascal-type residual construction in P^4. Starting from two quadruples of hyperplanes whose four diagonal intersection planes lie in a hyperplane, we show that the twelve residual planes lie on a cubic threefold. In the general case this cubic is the Segre cubic, and the construction recovers its fifteen planes and the associated Cremona--Richmond configuration. We also exhibit a point-line realization of this configuration in P^4 and show that it gives a (5,3)-geprofi set.

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arXiv2026

On the Jacobian algebras of Ziegler pairs of plane arrangements

We consider a Ziegler pair of plane arrangements, that is two plane arrangements $\mathcal{A}:f=0$ and $\mathcal{A}':f'=0$ in the projective space $\mathbb{P}^3$, such that the intersection lattices $L(\mathcal{A})$ and $L(\mathcal{A}')$ are isomorphic, but the Betti numbers of the minimal resolutions of their Jacobian algebras are not the same. We introduce several properties for such pairs and relate them to cones over Ziegler pairs of line arrangements in $\mathbb{P}^2$.

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arXiv2026

On combinatorial bounds for the total Tjurina numbers of certain curves and surfaces with isolated singularities

We investigate combinatorial bounds for the total Tjurina numbers of some plane curve arrangements. Focusing on arrangements of lines and conics in $\mathbb{P}^2$ that admit only ordinary quasi-homogeneous singularities, we derive new structural inequalities governing the distribution of multiple intersection points. As a consequence, we establish sharp lower bounds for the total Tjurina numbers of free line arrangements with bounded maximal multiplicity and, more generally, for free conic-line arrangements. In particular, we show that for a free arrangement of $d$ lines and $k$ conics, the total Tjurina number grows at least quadratically in $d$ and $k$, and we demonstrate that this bound is sharp. As an application of these planar results, we construct a special family of surfaces in $\mathbb{P}^{3}$ with only isolated singularities and arbitrarily large total Tjurina numbers. This provides new lower bounds for the total Tjurina numbers of certain hypersurfaces that are independent of detailed homological data.

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arXiv2025

On $\mathscr{M}$-arrangements of conics and lines with ordinary singularities

In this paper we study combinatorial aspects of reduced plane curves known as $\mathscr{M}$-curves. This notion is a natural generalization of maximizing plane curves, which are well-known in the theory of algebraic surfaces. We focus on $\mathscr{M}$-arrangements of conics and lines with ordinary singularities of multiplicity at most four. We provide numerical constraints on their existence, especially in terms of weak combinatorics, and then study in detail the case of arrangements consisting of one conic and lines. We also construct a new example with one conic and eleven lines, prove boundedness results for real arrangements of this type, and record a regularity consequence for the associated Milnor algebra and module of Jacobian syzygies.

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arXiv2025

Addition theorems for Ziegler pairs of hyperplane arrangements

Inspired by Terao's freeness conjecture, we examine Ziegler pairs, which are pairs of hyperplane arrangements that share the same underlying matroid but have different modules of logarithmic derivations. In this paper, we present a general construction that yields the first known families of Ziegler pairs in arbitrary dimension and size, starting from examples in the complex projective plane.

math.CO↗
arXiv2025

On plus-one generated arrangements of plane conics

In this paper, we examine the combinatorial properties of conic arrangements in the complex projective plane that possess certain quasi-homogeneous singularities. First, we introduce a new tool that enables us to characterize the property of being plus-one generated within the class of conic arrangements with some naturally chosen quasi-homogeneous singularities. Next, we present a classification result on plus-one generated conic arrangements admitting only nodes and tacnodes as singularities. Building on results regarding conic arrangements with nodes and tacnodes, we present new examples of strong Ziegler pairs of conic-line arrangements -- that is, arrangements having the same strong combinatorics but distinct derivation modules.

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arXiv2025

Free line arrangements with low maximal multiplicity

Let $\A$ be a free arrangement of $d$ lines in the complex projective plane, with exponents $d_1\leq d_2$. Let $m$ be the maximal multiplicity of points in $\A$. In this note, we describe first the simple cases $d_1 \leq m$. Then we study the case $d_1=m+1$, and describe which line arrangements can occur by deleting or adding a line to $\A$. When $d \leq 14$, there are only two free arrangements with $d_1=m+2$, namely one with degree $13$ and the other with degree $14$. We study their geometries in order to deepen our understanding of the structure of free line arrangements in general.

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arXiv2025

On homological properties of some Cynk-Szemberg octic hyperplane arrangements

In this paper we study Cynk-Szemberg octic hyperplane arrangements from the perspective of homological properties of their derivation modules. In particular, we define the notion of the type of hyperplane arrangements that will be used in our characterization of rigid Cynk-Szemberg octic hyperplane arrangements. Moreover, we deliver a combinatorial non-freeness criterion for essential hyperplane arrangements in $\mathbb{C}^{4}$.

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arXiv2024

On the geography of log-surfaces

This survey focuses on the geometric problem of log-surfaces, which are pairs consisting of a smooth projective surface and a reduced non-empty boundary divisor. In the first part, we focus on the geography problem for complex log-surfaces associated with pairs of the form $(\mathbb{P}^{2}, C)$, where $C$ is an arrangement of smooth plane curves admitting ordinary singularities. Specifically, we focus on the case in which $C$ is an arrangement consisting of smooth rational curves as its irreducible components. In the second part, containing original new results, we study log-surfaces constructed as pairs consisting of a complex projective $K3$ surface and a rational curve arrangement. In particular, we provide some combinatorial conditions for such pairs to have the log-Chern slope equal to $3$. Our survey is illustrated with many explicit examples of log-surfaces.

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arXiv2024

On Poincar\'e polynomials for plane curves with quasi-homogeneous singularities

We define a combinatorial object that can be associated with any conic-line arrangement with ordinary singularities, which we call the combinatorial Poincar\'e polynomial. We prove a Terao-type factorization statement on the splitting of such a polynomial over the rationals under the assumption that our conic-line arrangements are free and admit ordinary quasi-homogeneous singularities. Then we focus on the so-called $d$-arrangements in the plane. In particular, we provide a combinatorial constraint for free $d$-arrangements admitting ordinary quasi-homogeneous singularities.

math.AG↗
arXiv2024

A new hierarchy for complex plane curves

We define the type of a plane curve as the initial degree of the corresponding Bourbaki ideal. Then we show that this invariant behaves well with respect to the union of curves. Curves of type $0$ are precisely the free curves, while curves of type $1$ are the plus-one generated curves. In this paper, we first show that line arrangements and conic-line arrangements can exhibit all the theoretically possible types. In the second part, we study the properties of the curves of type $2$ and construct families of line arrangements and conic-line arrangements of this type.

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