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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.

14H50 Plane and space curves
14H25 Arithmetic ground fields for curves
14G15 Finite ground fields in algebraic geometry
Full Text: DOI
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[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
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[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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