Research Team:

• Piotr Pokora  (PI)

• Marek Janasz  (RI)

• Artur Bromboszcz  (PhD)

The main research goals:

• constructions of m-syzygy plane curves,

• constraints on weak-combinatorics of plane curves,

• Numerical Terao's Conjecture and various Ziegler pairs,

• plus-one generated curves,

• understanding the defect of plane curves,

• geometric applications of m-syzygy plane curves,

• monodromy of rational plane curves.

Research papers:

Published:

1. Singular plane curves: freeness and combinatorics (Michael Cuntz, Piotr Pokora), Innovations in Incidence Geometry - Algebraic, Topological and Combinatorial 22(1): 47 - 65 (2025), Green Access AAM.

2. On symmetric plane quartics (Marek Janasz), Taiwanese Journal of Mathematics 29(1): 1 - 11 (2025), Open Access with AAM.

3. Minimal plus-one generated line arrangements with double and triple intersection points (Artur Bromboszcz), International Journal of Mathematics Vol. 36, No. 11: Art Id. 2550040 (11 pages) (2025), Green Access AAM.

4. On the existence of maximizing curves of odd degrees (Marek Janasz, Izabela Leśniak), Proceedings of the American Mathematical Society 153: 4633 - 4642 (2025), Green Access AAM.

5. On Poincare polynomials for plane curves with quasi-homogeneous singularities (Piotr Pokora), Bulletin of the London Mathematical Society 57(8): 2549 - 2560 (2025), Green Access AAM.

6. On the addition technique for Betti and Poincare polynomials of plane curves (Piotr Pokora), Bulletin of the Polish Academy of Sciences - Mathematics 72(2): 111 - 118 (2024), Green Access AAM.

7. On free line arrangements with double, triple and quadruple points (Marek Janasz, Izabela Leśniak), European Journal of Mathematics 11: Art Id. 81 (14 pages) (2025), Open Access.

8. Defect of irreducible plane curves with simple singularities (Piotr Pokora), Annales Polonici Mathematici (Special Volume dedicated to A. Płoski) 135: 295 - 303 (2025), Green Access AAM.

9. On plus-one generated arrangements of conics (Artur Bromboszcz, Bartosz Jarosławski, Piotr Pokora), Geometriae Dedicata 220: Art Id. 10, 15 pages (2026), Open Access.

10. Free line arrangements with low maximal multiplicity (Alexandru Dimca, Lukas Kuehne, Piotr Pokora), Journal of Algebraic Combinatorics 63(3): Art. Id. 45, 19 pages (2026), Green Access AAM.

11. On homological properties of some Cynk-Szemberg octic hyperplane arrangements (Marek Janasz, Piotr Pokora), Journal of the Korean Mathematical Society (2026), Open Access.

Early View/Accepted:

12. On combinatorics of plus-one generated line arrangements (Artur Bromboszcz), Bollettino dell'Unione Matematica Italiana, https://doi.org/10.1007/s40574-025-00517-z (2026), Open Access.

13. A new hierarchy for complex plane curves (Takuro Abe, Alex Dimca, Piotr Pokora), Canadian Mathematical Bulletin, https://doi.org/10.4153/S0008439525101422 (2026), Green Access AAM.

14. On the Numerical Terao's Conjecture and Ziegler pairs for line arrangements (Dante Luber, Lukas Kuhne, Piotr Pokora), Journal of Commutative Algebra (2026), Green Access AAM.

15. Constructions of curve arrangements using point-line operators (Piotr Pokora, Xavier Roulleau), to appear in Experimental Mathematics (202X), Green Access AAM.

16. On the geography of log-surfaces (Bartosz Naskręcki, Piotr Pokora), to appear in EMS Surveys in Mathematical Sciences (202X), Green Access AAM.

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Under Review:

17. Addition theorems for Ziegler pairs of hyperplane arrangements (Takuro Abe, Lukas Kuhne, Piotr Pokora), arXiv:2509.19011, under review.

18. Unexpected hypersurfaces of type (d+k,d) (Marek Janasz, Grzegorz Malara, Halszka Tutaj-Gasińska), arXiv:2511.10772, under review.

19. M -arrangements of conics and lines with ordinary singularities (Marek Janasz, Piotr Pokora), arXiv:2512.24707, under review.

20. On the boundedness of some real line arrangements of type at most one (Marek Janasz), arXiv:2602.05409, under review.

21. On combinatorial bounds for the total Tjurina numbers of certain curves and surfaces with isolated singularities (Piotr Pokora), arXiv:2602.21982, under review.

22. On the Jacobian algebras of Ziegler pairs of plane arrangements (Alex Dimca, Piotr Pokora), arXiv:2604.25637, under review.

Conferences:

• Varieties with Boundries, 25 - 28 June 2024 Wakkanai (Japan), Invited Speaker - Piotr Pokora: "On the geography problem of log-surfaces".

• XXIV Coloquio Latinoamericano de Álgebra, 22-26 July 2024, Santiago (Chile), Invited Speaker Algebraic Geometry Session - Piotr Pokora: "Rational curve arrangements: geometry and combinatorics".

• 9. Forum Matematyków Polskich, 9 - 13 września 2024, Katowice, GKLW session Invited Speaker - Piotr Pokora: "Defect of irreducible plane curves with simple singularities".

• XLV Konferencja i Warsztaty "Geometria Analityczna i Algebraiczna", Łódź, 6 - 10 January 2025 - Artur Bromboszcz: "Combinatorial properties of plus-one generated line arrangements" and Piotr Pokora: "Poincare polynomials for curve arrangements".

• Geometry of Arrangements Mini-Workshop, Hannover, 20 - 21 March 2025, Invited Speaker - Piotr Pokora: "Poincare polynomials of plane curves with quasi-homogeneous singularities.

• New Perspectives on Hyperplane Arrangements, Bochum, 12 - 16 May 2025, Invited Speaker - Piotr Pokora: "A new hierarchy for line arrangements".

• Summer Institute School Algebraic Geometry, Fort Collins (Colorado USA), 14 - 18 July 2025, Invited Contributed Talk Speaker - Piotr Pokora: "Free curves in algebraic geometry".

• XLVI Conference and Worskhop "Analytic and Algebraic Geometry", Łódź 7 - 10 January 2026, Talk: M-arrangements of conics and lines.

• 77th BMC, 30th March - 2nd April, Cardiff (Wales) - Invited Geometry Session Speaker - Piotr Pokora: "Curves and surfaces with extreme (homological) properties".

External Seminars:

• Moduli and Friends ZOOM Seminar, 20th May 2024 (Bucharest): Talk: Constructions of free arrangements using point-line operators (Piotr Pokora).

• Combinatorics and Interactions IMPAN Kraków, 16.12.2025. Talk: Interactions between algebra and combinatorics: variations on Terao's freeness conjecture (Piotr Pokora).

• Algebra Seminar at Jagiellonian University in Kraków, 12.03.2026. Talk: On homological types of hypersurfaces (Piotr Pokora).

• 77th BMC, 30th March - 2nd April, Cardiff - Invited Geometry Session Speaker, Talk "Curves and surfaces with extreme (homological) properties".