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Nonsingular plane filling curves of minimum degree over a finite field and their automorphism groups: Supplements to a work of Tallini. (English) Zbl 1259.14023
Let \(\mathbb{F}_q\) be a finite field. The authors find nonsingular projective plane curves over \(\mathbb{F}_q\), of degree \(q+2\), whose \(\mathbb{F}_q\)-rational points fill the whole projective plane \(\mathbb{P}^2(\mathbb{F}_q)\). The degree \(q+2\) is minimal for a curve with this property. Let \(x,y,z\) be homogeneous coordinates of \(\mathbb{P}^2\), and \(U=y^qz-yz^q\), \(V=z^qx-zx^q\), \(W=x^qy-xy^q\). The curves have equations \(F_A=0\), where \(F_A=(x\;y\;z)A(U\;V\;W)^t\), for a \(3\times 3\) matrix \(A\) with entries in \(\mathbb{F}_q\), whose characteristic polynomial is irreducible. The automorphism groups of these curves are also determined.
The results of the paper are similar to those obtained by G. Tallini in [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 30, 706–712 (1961; Zbl 0107.38104), Rend. Mat. Appl., V. Ser. 20, 431–479 (1961; Zbl 0106.35604)].

MSC:
14G15 Finite ground fields in algebraic geometry
14H37 Automorphisms of curves
14H50 Plane and space curves
14G05 Rational points
11G20 Curves over finite and local fields
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