Some open questions about line arrangements in the projective plane

• Urzúa, Giancarlo ^[1]
  1. [1] Pontificia Universidad Católica de Chile

     Pontificia Universidad Católica de Chile

     Santiago, Chile

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 41, Nº. Extra 2, 2022 (Ejemplar dedicado a: Special Issue on Open Questions in Geometry), págs. 517-536
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-5277
• Enlaces
  • Texto completo
• Resumen

  • Despite that the study of line arrangements in the projective plane is old and elemental, there is still a long list of intriguing open questions and applications to modern mathematics. Our goal is to discuss part of that list, focusing on the connection with Chern invariants and pointing towards configurations of rational curves.

• Referencias bibliográficas
  • M. Aigner and G. M. Ziegler, Proofs from THE BOOK, 3th ed. Berlin: Springer, 2004.
  • G. Barthel, F. Hirzebruch, and T. Höfer, Geradenkonfigurationen und Algebraische Flächen. Braunschweig: Teubner, 1987. https://doi.org/10.1007/978-3-322-92886-3
  • A. Bassa, and A. U. O. Kisisel, “The Only Complex 4-Net Is the Hesse Configuration”, 2020, arXiv: 2002.02660v2.
  • T. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Lundman, P. Pokora, and T. Szemberg, “Bounded negativity and arrangements of lines”,...
  • J. I. Cogolludo-Agustín, Topological invariants of the complement to arrangements of rational plane curves. Providence: AMS, 2002. https://doi.org/10.1090/memo/0756
  • M. Cuntz, “Minimal fields of definition for simplicial arrangements in the real projective plane”, Innovations in incidence geometry: algebraic,...
  • M. Cuntz, “Simplicial arrangements with up to 27 lines”, Discrete & computational geometry, vol. 48, no. 3, pp. 682–701, 2012. https://doi.org/10.1007/s00454-012-9423-7
  • N. G. de Bruijn and P. Erdös, “On a combinatorial problem”, Proceedings of the Section of Sciences of the Koninklijke Nederlandse Akademie...
  • A. Dimca, B. Harbourne, and G. Sticlaru, “On the bounded negativity conjecture and singular plane curves”, 2021, arXiv: 2101.07187v3.
  • A. Dimca, M. Janasz, and P. Pokora, “On plane conic arrangements with nodes and tacnodes”, 2022, arXiv: 2108.04004v4.
  • A. Dimca and P. Pokora, “On conic-line arrangements with nodes, tacnodes, and ordinary triple points”, 2022, arXiv: 2111.12349v2.
  • I. Dolgachev, “Abstract configurations in algebraic geometry”, 2003, arXiv:math/0304258v1.
  • I. Dolgachev, A. Laface, U. Persson, and G. Urzúa, “Chilean configuration of conics, lines and points”, 2020, arXiv: 2008.09627v1.
  • M. Dumnicki, Ł. Farnik, A. Główka, M. Lampa-Baczyńska, G. Malara, T. Szemberg, J. Szpond, and H. Tutaj-Gasińska, “Line arrangements with the...
  • S. Eterović, F. Figueroa, and G. Urzúa, “On the geography of line arrangements”, 2019, arXiv: 1805.00990v2.
  • S. Eterović, “Logarithmic Chern slopes of arrangements of rational sections in Hirzebruch surfaces”, Master Thesis, Pontificia Universidad...
  • B. Grünbaum, “A catalogue of simplicial arrangements in the real projective plane”, Ars mathematica contemporanea, vol. 2, no. 1, pp. 1–25,...
  • B. Grünbaum, Configurations of points and lines. Providence: AMS, 2009.
  • F. Hao, “Weak bounded negativity conjecture”, Proceedings of the American mathematical society, vol. 147, no. 8, pp. 3233–3238, 2019. https://doi.org/10.1090/proc/14376
  • B. Harbourne, “Asymptotics of linear systems, with connections to line arrangements”, 2017, arXiv: 1705.09946v1
  • D. Hilbert and S. Cohn-Vossen, Geometry and the imagination Chelsea. New York: Publishing Company, 1952.
  • F. Hirzebruch, “Arrangements of lines and algebraic surfaces”, in Arithmetic and geometry, vol. 2, M. Artin and J. Tate, Eds. Boston: Birkhäuser,...
  • F. Hirzebruch, “Singularities of algebraic surfaces and characteristic numbers”, in The Lefschetz centennial Conference, vol. 1, D. Sundararaman,...
  • D. J. Houck and M. E. Paul, “On a theorem of De Bruijn and Erdös”, Linear algebra and its applications, vol. 23, pp. 157–165, 1979. https://doi.org/10.1016/0024-3795(79)90099-5
  • B. Hunt, “Complex manifold geography in dimension 2 and 3”, Journal of differential geometry, vol. 30, no. 1, pp. 51-153, 1989. https://doi.org/10.4310/jdg/1214443287
  • S. Iitaka, “Geometry on complements of lines in P²”, Tokyo journal of mathematics, vol. 1, no. 1, pp. 1-19, 1978. https://doi.org/10.3836/tjm/1270216590
  • Y. J. Ionin and M. S. Shrikhande, Combinatorics of symmetric designs. Cambridge: Cambridge University, 2006. https://doi.org/10.1017/CBO9780511542992
  • S. Jukna, Extremal combinatorics, 2nd ed. Berlin: Springer, 2011. Https://doi.org/10.1007/978-3-642-17364-6
  • D. Kohel, X. Roulleau, and A. Sarti, “A special configuration of 12 conics and generalized Kummer surfaces”, 2021, arXiv: 2004.11421v2
  • J. Kollár, “Is there a topological Bogomolov-Miyaoka-Yau inequality?”, Pure and applied mathematics quarterly, vol. 4, no. 2, pp. 203–236,...
  • G. Korchmáros, G. P. Nagy and N. Pace, “3-nets realizing a group in a projective plane”, Journal of algebraic combinatorics, vol. 39, no....
  • G. Korchmáros, G. P. Nagy, and N. Pace, “K-nets embedded in a projective plane over a field”, Combinatorica, vol. 35, no. 1, pp. 63–74, 2015....
  • A. Langer, “Logarithmic orbifold euler numbers of surfaces with applications”, Proceedings of the London mathematical society, vol. 86, no....
  • Y. Lee and J. Park, “A simply connected surface of general type with pg= 0 and K²= 2”, Inventiones mathematicae, vol. 170, no. 3,...
  • S. H. Lee and R. Vakil, “Mnëv-Sturmfels universality for schemes”, in A celebration of algebraic geometry, B. Hassett, J. McKernan, J. Starr...
  • G. Megyesi, “Configurations of conics with many tacnodes”, Tohoku mathematical journal, vol. 52, no. 4, pp. 555-577, 2000. https://doi.org/10.2748/tmj/1178207755
  • G. P. Nagy and N. Pace, “On small 3-nets embedded in a projective plane over a field”, Journal of combinatorial theory, Series A, vol. 120,...
  • P. Orlik and L. Solomon, “Arrangements defined by unitary reflection groups”, Mathematische annalen, vol. 261, no. 3, pp. 339–357, 1982. https://doi.org/10.1007/bf01455455
  • H. Park, J. Park, and D. Shin, “A simply connected surface of general type type with pg= 0 and K²= 3”, Geometry & topology, vol....
  • H. Park, J. Park, and D. Shin, “A simply connected surface of general type with pg= 0 and K²= 4”, Geometry & topology, vol. 13,...
  • U. Persson, “An introduction to the geography of surfaces of general type”, Proceedings of Symposia in Pure Mathematics, vol. 46, pp. 195–218,...
  • J. V. Pereira and S. Yuzvinsky, “Completely reducible hypersurfaces in a pencil”, Advances in mathematics, vol. 219, no. 2, pp. 672–688, 2008....
  • P. Pokora and T. Szemberg, “Conic-line arrangements in the complex projective plane”, 2022, arXiv: 2002.01760v3.
  • J. Reyes, G. Urzúa, “Rational configurations in K3 surfaces and simply-connected pg = 1 surfaces for K² = 1, 2, 3, 4, 5, 6, 7, 8,...
  • X. Roulleau, “Conic configurations via dual of quartic curves”, 2020, arXiv: 2002.05681v3.
  • X. Roulleau and G. Urzúa, “Chern slopes of simply connected complex surfaces of general type are dense in [2,3]”, Annals of mathematics, vol....
  • D. Ruberman and L. Starkston, “Topological realizations of line arrangements”, International mathematics research notices, vol. 2019, no....
  • H. Schenck and Ş. Tohǎneanu, “Freeness of conic-line arrangements in ℙ²”, Commentarii mathematici helvetici, pp. 235–258, 2009. https://doi.org/10.4171/cmh/161
  • I. N. Shnurnikov, “A tK inequality for arrangements of pseudolines”, Discrete & computational geometry, vol. 55, no. 2, pp. 284–295, 2016....
  • A. J. Sommese, “On the density of ratios of Chern numbers of algebraic surfaces”, Mathematische annalen, vol. 268, no. 2, pp. 207–221, 1984....
  • A. Stern and G. Urzúa, “KSBA surfaces with elliptic quotient singularities, π₁ = 1, pg = 0, and K² = 1, 2”, Israel journal of...
  • J. Stipins, “Old and new examples of k-nets in ℙ²”, 2007, arXiv: math/0701046v1.
  • G. Tian and S. T. Yau, “Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry”, in...
  • P. Tretkoff, Complex ball quotients and line arrangements in the projective plane (MN-51). Princeton: Princenton University, 2016. https://doi.org/10.23943/princeton/9780691144771.001.0001
  • G. Urzúa, “Arrangements of curves and algebraic surfaces”, Ph.D. Thesis, University of Michigan, 2008.
  • G. Urzúa, “On line arrangements with applications to 3-nets”, Advances in geometry, vol. 10, no. 2, pp. 287–310, 2010. https://doi.org/10.1515/advgeom.2010.006
  • G. Urzúa, “Arrangements of curves and algebraic surfaces”, Journal of algebraic geometry, vol. 19, no. 2, pp. 335–365, 2009. https://doi.org/10.1090/s1056-3911-09-00520-7
  • G. Urzúa, “Arrangements of rational sections over curves and the varieties they define”, Rendiconti lincei - matematica e applicazioni, vol....
  • R. Vakil, “Murphy’s law in algebraic geometry: Badly-behaved deformation spaces”, Inventiones mathematicae, vol. 164, no. 3, pp. 569–590,...
  • S. Yuzvinsky, “Realization of finite Abelian groups by nets in ℙ²”, Compositio mathematica, vol. 140, no. 06, pp. 1614–1624, 2004. https://doi.org/10.1112/s0010437x04000600