Piotr Pokora

Lifting negative curves from finite fields and Seshadri constants at ten points

Piotr Pokora, Tomasz Szemberg

We use finite-field calculations and mixed-characteristic deformation to construct nef divisors on very general complex blow-ups of the plane. A lifting and smoothing criterion con…

A generalized Terao Conjecture for line arrangements

Alexandru Dimca, Piotr Pokora

For a line arrangement $A : f = 0$ in $P^{2}$ , let $ν (A)$ be the maximal dimension of a graded piece of the Jacobian module of $f$ . We study the conje…

Demailly's Inequality for Finite Sets of Points in Positive Characteristic

Piotr Pokora, Tomasz Szemberg

Let $k$ be an algebraically closed field of characteristic $p > 0$ , $n \geq 1$ , and let $X \subset P_{k}^{n}$ be a finite nonempty set of distinct points with the defining ideal $I=I…

On the Numerical Terao Conjecture

Piotr Pokora

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 quas…

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

Alexandru Dimca, Piotr Pokora

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 comm…

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

Mateusz Michałek, Piotr Pokora

We record an explicit counterexample to a refined form of Purdy's inequality for essential hyperplane arrangements in projective three-space. Let $A$ be an arrangement of…

A $(2 4_{4}, 3 2_{3})$ -configuration on the Schur quartic with logarithmic Chern slope $14/5$

Bartosz Naskręcki, Piotr Pokora

Let $X \subset P^{3}$ 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 $Q (- 3)$ …

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

Piotr Pokora, Tomasz Szemberg

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 twe…

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

Alexandru Dimca, Piotr Pokora

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 rel…

On the Jacobian algebras of Ziegler pairs of plane arrangements

Alexandru Dimca, Piotr Pokora

We consider a Ziegler pair of plane arrangements, that is two plane arrangements $A : f = 0$ and $A^{'} : f^{'} = 0$ in the projective space $P^{3}$ , such that the int…

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

Piotr Pokora

We investigate combinatorial bounds for the total Tjurina numbers of some plane curve arrangements. Focusing on arrangements of lines and conics in $P^{2}$ that admit only o…

On $M$ -arrangements of conics and lines with ordinary singularities

Marek Janasz, Piotr Pokora

In this paper we study combinatorial aspects of reduced plane curves known as $M$ -curves. This notion is a natural generalization of maximizing plane curves, which are we…

On plus-one generated arrangements of plane conics

Artur Bromboszcz, Bartosz Jarosławski, Piotr Pokora

In this paper, we examine the combinatorial properties of conic arrangements in the complex projective plane that possess certain quasi-homogeneous singularities. First, we introdu…

Free line arrangements with low maximal multiplicity

Alexandru Dimca, Lukas Kühne, Piotr Pokora

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 d…

On homological properties of some Cynk-Szemberg octic hyperplane arrangements

Marek Janasz, Piotr Pokora

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 o…

On the addition technique for Betti and Poincaré polynomials of plane curves

Piotr Pokora

Using the addition technique, we present polynomial identities for the Betti and Poincaré polynomials of reduced plane curves.

On the geography of log-surfaces

Bartosz Naskręcki, Piotr Pokora

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,…

math.AG2024★ 3 cited

On Poincaré polynomials for plane curves with quasi-homogeneous singularities

Piotr Pokora

We define a combinatorial object that can be associated with any conic-line arrangement with ordinary singularities, which we call the combinatorial Poincaré polynomial. We prove a…

A new hierarchy for complex plane curves

Takuro Abe, Alexandru Dimca, Piotr Pokora

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. Cu…

math.AG2024★ 1 cited

On the numerical Terao's conjecture and Ziegler pairs for line arrangements

Lukas Kühne, Dante Luber, Piotr Pokora

In this paper we present a smallest possible counterexample to the Numerical Terao's Conjecture in the class of line arrangements in the complex projective plane. Our example consi…

Defect of irreducible plane curves with simple singularities

Piotr Pokora

In this note we focus on the defect of singular plane curve that was recently introduced by Dimca. Roughly speaking, the defect of a reduced plane curve measures the discrepancy fr…