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Curves containing all points of a finite projective Galois plane. (English) Zbl 1428.14052
Summary: In the projective plane \(P G(2, q)\) over a finite field of order \(q\), a Tallini curve is a plane irreducible (algebraic) curve of (minimum) degree \(q + 2\) containing all points of \(P G(2, q)\). Such curves were investigated by G. Tallini [Rend. Mat. Appl., V. Ser. 20, 431–479 (1961; Zbl 0106.35604); Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 30, 706–712 (1961; Zbl 0107.38104)], and by M. Homma and S. J. Kim [Linear Algebra Appl. 438, No. 3, 969–985 (2013; Zbl 1259.14023)]. Our results concern the automorphism groups, the Weierstrass semigroups, the Hasse-Witt invariants, and quotient curves of the Tallini curves.

MSC:
14H50 Plane and space curves
14H25 Arithmetic ground fields for curves
14G15 Finite ground fields in algebraic geometry
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[1] Abatangelo, V.; Korchmáros, G., Irreducible hypersurfaces of minimal degree containing all points of a finite projective space over a finite field, (Trends in Incidence and Galois Geometries: a Tribute to Giuseppe Tallini, Quad. Mat., vol. 19, (2009), Dept. Math., Seconda Univ. Napoli Caserta), 1-17 · Zbl 1235.51011
[2] Cossidente, A.; Siciliano, A., Plane algebraic curves with Singer automorphisms, J. Number Theory, 99, 373-382, (2003) · Zbl 1074.14025
[3] Gorenstein, D., An arithmetic theory of adjoint plane curves, Trans. Am. Math. Soc., 72, 414-436, (1952) · Zbl 0046.38503
[4] Hirschfeld, J. W.P.; Korchmáros, G.; Torres, F., Algebraic curves over a finite field, (2008), Princeton Univ. Press Princeton and Oxford, xx+696 pp · Zbl 1200.11042
[5] Homma, M.; Kim, S. J., Nonsingular plane filling curves of minimum degree over a finite field and their automorphism groups: supplements to a work of tallini, Linear Algebra Appl., 438, 969-985, (2013) · Zbl 1259.14023
[6] M. Montanucci, P. Speziali, On the a-number of curves of Fermat and Hurwitz type, preprint, 2016. · Zbl 1427.11059
[7] Pellikaan, R., The Klein quartic, the Fano plane and curves representing designs, (Codes, Curves, and Signals, Urbana, IL, 1997, Kluwer Int. Ser. Eng. Comput. Sci., vol. 485, (1998), Kluwer Acad. Publ. Boston, MA), 9-20 · Zbl 1009.94016
[8] Tallini, G., Sulle ipersuperfici irriducibili d’ordine minimo che contengono tutti i punti di uno spazio di Galois \(S_{r, q}\), Rend. Mat. Appl. (5), 20, 431-479, (1961) · Zbl 0106.35604
[9] Tallini, G., Le ipersuperficie irriducibili d’ordine minimo che invadono uno spazio di Galois, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (8), 30, 706-712, (1961) · Zbl 0107.38104
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