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Piotr Pokora

Publications and source records attributed to Piotr Pokora.

At least 19 recordsLinked to original sources
arXiv2026

Seshadri constants of singular loci of line arrangements with at most twelve lines

Let ${\mathcal L}$ be an arrangement of $d$ lines in the complex projective plane and let $Z={\operatorname{Sing}}({\mathcal L})$. A question of Pokora asks whether the multipoint Seshadri constant of ${\mathcal O}_{\mathbb{P}^2}(1)$ at $Z$ is always computed by a line of the arrangement containing the largest number of points of $Z$. We prove that the answer is affirmative for every arrangement of at most twelve lines. The proof uses a Bézout-type covering criterion, elementary restrictions on the intersection lattice, and a detailed analysis of the two extremal combinatorial types that can occur for twelve lines. One type is the Hesse arrangement; it is handled using a net of quartics through its twelve double points. The other type would yield a uniquely determined incidence structure on thirteen lines. We show that this incidence structure is not realizable over any field.

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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 $\mathbb{P}^2$.

math.AG↗
arXiv2026

There are no Miyaoka--Kobayashi octics

Let $C$ be a reduced plane curve of degree $n$ with only ADE singularities. Using Kähler-Einstein metrics, Kobayashi proved an upper bound, in terms of $n$, for a sum of local invariants of the singular points of $C$. For even $n$ this bound had been obtained earlier by Hirzebruch and Ivinskis. Curves of even degree for which equality holds are called Miyaoka-Kobayashi curves, and the only known examples are three types of plane sextics. We show that there are no Miyaoka--Kobayashi curves of degree $8$, and additionally that a Miyaoka-Kobayashi curve of degree $10$ cannot be free. The key new ingredient is a new linear constraint on the singularities of such curves, obtained by differentiating Langer's orbifold Bogomolov-Miyaoka-Yau inequality.

math.AG↗
arXiv2026

Containment problems for fat point schemes via asymptotic multiplier ideals

We study a multiplier-ideal approach to the containment problem for radical ideals of points in projective space. The expected asymptotic multiplier-ideal computation depends on controlling residual base loci of the relevant symbolic linear systems. We isolate this issue in the notion of clean asymptotic multiplier ideals. Under this hypothesis, a failure of \[ I^{(2r-1)}\not\subseteq I^r \] forces \[ 2r\widehatα(I)\geq α(I^{(2r-1)})+3, \] and in particular $\widehatα(I)\geq 3$. We show that the dual-Hesse configuration is clean although its symbolic systems have a genuinely non-trivial residual asymptotic base part. We also verify cleanliness for star configurations of lines, giving a further infinite family of non-monomial examples, and formulate a cleanliness conjecture for singular loci of line arrangements.

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arXiv2026

Topological inequalities for conic-line arrangements with some prescribed singularities

We study arrangements of conics and lines in the complex projective plane from the viewpoint of 4-manifolds. We introduce combinatorial conic-line arrangements, the conic-line counterpart of rank-3 matroids, together with their topological and smooth realisations, and we ask which restrictions on a complex arrangement survive when its components are replaced by locally-flat spheres in the same homology classes with the same local singularity models. For an ADE combinatorial conic-line arrangement of even total degree admitting a topological realisation we prove a Hirzebruch-type inequality; the proof uses only branched covers and the local topology of rational double points. In the complex-algebraic category we show that equality holds exactly when the arrangement is a maximising curve, hence free with prescribed exponents. We then introduce divisible and odd combinatorial conic-line arrangements and obtain inequalities and congruences for the multiplicity sequence of their resolutions, via cyclic branched covers, the G-signature theorem, spin structures, and Furuta's 10/8-theorem.

math.GT↗
arXiv2026

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

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 converts a reducible nodal divisor over $\mathbb{F}_{127}$ into a smooth genus-$10$ curve of class $117H-37(E_1+\cdots+E_{10})$, retaining control of its first-order obstruction map. Its normal bundle in a suitable deformation is semistable, so Petrakiev's multiplicity estimate yields the nef class $27379H-8658(E_1+\cdots+E_{10})$ at ten very general points. Their multipoint Seshadri constant is therefore at least $8658/27379$, improving Petrakiev's bound $228/721$. The geometric input is an existence criterion in characteristic zero verified by computations in positive characteristic.

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arXiv2026

A generalized Terao Conjecture for line arrangements

For a line arrangement $\mathcal{A} \colon f=0$ in $\mathbb{P}^2$, let $ν(\mathcal{A})$ be the maximal dimension of a graded piece of the Jacobian module of $f$. We study the conjecture that $ν(\mathcal{A})$ depends only on the intersection lattice of $\mathcal{A}$. It is known that $\mathcal{A}$ is free if and only if $ν(\mathcal{A})=0$, and hence this conjecture is a strengthening of Terao's conjecture for line arrangements. We show that the conjecture holds for arrangements of at most $13$ lines, except possibly for arrangements of exactly $13$ lines whose intersection points have maximal multiplicity $5$.

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arXiv2026

Construction of free arrangements using point-line operators

We construct new examples of free curve arrangements in the complex projective plane using point-line operators recently defined by the second author. In particular, we construct a new example of a conic-line arrangement with ordinary quasi-homogeneous singularities that has non-trivial monodromy.

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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. 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.

math.AG↗
arXiv2026

On the Alexander polynomials of conic-line arrangements

In the present paper we compute Alexander polynomials for certain classes of conic-line arrangements in the complex projective plane which are related to pencils. We prove two general results for curve arrangements coming from Halphen pencils of index $k\geq 2$. Then we apply them to the Hesse arrangement of conics and to some of its degenerations. The results are completed by computations using computer algebra. In particular, we construct conic-line arrangements which are non-reduced pencil-type arrangements and have as roots of their Alexander polynomials roots of unity of order 7. Such roots are not known and are conjectured not to exist in the class of line arrangements.

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arXiv2026

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

Let $k$ be an algebraically closed field of characteristic $p>0$, $n\ge1$, and let $X\subset \mathbb P^n_k$ be a finite nonempty set of distinct points with the defining ideal $I=I(X)$. We give a proof of Demailly's inequality in positive characteristic. The argument is based on the Frobenius--Hasse derivative method used in this context by Hà and Sivakumar. The key additional observation is a strict-growth lemma for the initial degrees of symbolic powers of a finite set of affine points: $$ α(J^{(t)})\ge α(J^{(t-1)})+1\qquad(t\ge1). $$ In characteristic $p$, if a minimum-degree polynomial has a nonzero first ordinary derivative, this follows by differentiation; if all first ordinary derivatives vanish, perfectness gives a $p$th root and an induction on the symbolic exponent. The remainder of the proof uses the $q=p^e$ Frobenius decomposition and a maximal Hasse derivative.

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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 Grünbaum-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 Grünbaum-Shephard defects for these arrangements.

math.CO↗
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.

math.AG↗
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 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.

math.AG↗
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.

math.AG↗
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 $π:Y\to X$ blows up the triple points and $B=(π^{-1}D)_{\mathrm{red}}$, 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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