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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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