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Equivalence classes of Latin squares and nets in P 2
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 2, pp. 335-351.

The fundamental combinatorial structure of a net in P 2 is its associated set of mutually orthogonal Latin squares. We define equivalence classes of sets of orthogonal Latin squares by label equivalences of the lines of the corresponding net in P 2 . Then we count these equivalence classes for small cases. Finally we prove that the realization spaces of these classes in P 2 are empty to show some non-existence results for 4-nets in P 2 .

La structure combinatoire fondamentale d’un filet dans P 2 est donnée par l’ensemble des carrés latins orthogonaux associé. Nous définissons des classes d’équivalence de carrés latins orthogonaux a l’aide de classes d’équivalence des lignes apparaisant dans le filet de P 2 . Nous comptons le nombre de classes d’équivalence pour certains exemples de carrés petits. Finalement, nous montrons que les espaces de réalisations de ces classes pour n=4 et k=4,5,6 sont vides et nous en déduisons que les filets correspondants n’existent pas.

Published online:
DOI: 10.5802/afst.1409
@article{AFST_2014_6_23_2_335_0,
     author = {Corey Dunn and Matthew Miller and Max Wakefield and Sebastian Zwicknagl},
     title = {Equivalence classes of {Latin} squares and nets in ${\mathbb{C}P}^2$},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {335--351},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 23},
     number = {2},
     year = {2014},
     doi = {10.5802/afst.1409},
     zbl = {1296.05030},
     mrnumber = {3205596},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1409/}
}
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Corey Dunn; Matthew Miller; Max Wakefield; Sebastian Zwicknagl. Equivalence classes of Latin squares and nets in ${\mathbb{C}P}^2$. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 2, pp. 335-351. doi : 10.5802/afst.1409. https://afst.centre-mersenne.org/articles/10.5802/afst.1409/

[1] Julian (R.), Abel (R.), Colbourn (C. J.), Wojtas (M.).— Concerning seven and eight mutually orthogonal Latin squares, J. Combin. Des. 12, no. 2, p. 123-131 (2004). | MR: 2036650 | Zbl: 1033.05018

[2] Bartolo (E. A.), Cogolludo-Agustin (J.I.), Libgober (A.).— Depth of characters of curve complements and orbifold pencils (2011).

[3] Bartolo (E. A.), Cogolludo-Agustin (J.I.), Libgober (A.).— Depth of cohomology support loci for quasi-projective varieties via orbifold pencils (2012).

[4] Björner (A.), Las Vergnas (M.), Sturmfels (B.), White (N.), Ziegler (G. N.).— Oriented matroids, second ed., Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge (1999). | MR: 1744046 | Zbl: 0944.52006

[5] Brualdi (R. A.).— Introductory combinatorics, fifth ed., Pearson Prentice Hall, Upper Saddle River, NJ (2010). | MR: 2655770 | Zbl: 0915.05001

[6] Chowla (S.), Erdös (P.), Straus (E. G.).— On the maximal number of pairwise orthogonal Latin squares of a given order, Canad. J. Math. 12, p. 204-208 (1960). | MR: 122730 | Zbl: 0093.32001

[7] Colbourn (C. J.), Dinitz (J. D.) (eds.).— The CRC handbook of combinatorial designs, CRC Press Series on Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, (1996). | Zbl: 0836.00010

[8] Dénes (J.), Keedwell (A. D.).— Latin squares and their applications, Academic Press, New York (1974). | MR: 351850 | Zbl: 0754.05018

[9] Denham (G.), Suciu (A. I.).— Multinets, parallel connections, and milnor fibrations of arrangements, (2012).

[10] Falk (M.), Yuzvinsky (S.).— Multinets, resonance varieties, and pencils of plane curves, Compos. Math. 143, no. 4, p. 1069-1088 (2007). | MR: 2339840 | Zbl: 1122.52009

[11] Kawahara (Y.).— The non-vanishing cohomology of Orlik-Solomon algebras, Tokyo J. Math. 30, no. 1, 223-238 (2007). | MR: 2328065 | Zbl: 1132.52027

[12] Libgober (A.), Yuzvinsky (S.).— Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 121, no. 3, p. 337-361 (2000). | MR: 1761630 | Zbl: 0952.52020

[13] Orlik (P.), Terao (H.).— Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. | MR: 1217488 | Zbl: 0757.55001

[14] Pereira (J. V.), Yuzvinsky (S.).— Completely reducible hypersurfaces in a pencil, Adv. Math. 219, no. 2, p. 672-688 (2008). | MR: 2435653 | Zbl: 1146.14005

[15] Reidemeister (K.).— Topologische Fragen der Differentialgeometrie. V. Gewebe und Gruppen, Math. Z. 29, no. 1, p. 427-435 (1929). | EuDML: 168080 | MR: 1545014

[16] Stipins (J.).— III, On finite k-nets in the complex projective plane, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.) University of Michigan (2007). | MR: 2710620

[17] Tang (Y. L.), Liang Li (Q.).— A new upper bound for the largest number of mutually orthogonal Latin squares, Math. Theory Appl. (Changsha) 25, no. 3, p. 60-63 (2005). | MR: 2204935

[18] Tarry (G.).— Le problème de 36 officeurs, Compte Rendu de l’Association Française pour l’Avancement de Science Naturel, no. 2, p. 170-203 (1901). | JFM: 32.0219.04

[19] Urzua (G. A.).— On line arrangements with applications to 3-nets, 04 (2007). | Zbl: 1191.14066

[20] Urzua (G. A.).— Arrangements of curves and algebraic surfaces, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.) University of Michigan (2008). | MR: 2712257

[21] Yuzvinsky (S.).— Realization of finite abelian groups by nets in 2 , Compos. Math. 140, no. 6, p. 1614-1624 (2004). | MR: 2098405 | Zbl: 1066.52027

[22] Yuzvinsky (S.).— A new bound on the number of special fibers in a pencil of curves, Proc. Amer. Math. Soc. 137, no. 5, p. 1641-1648 (2009). | MR: 2470822 | Zbl: 1173.14021

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