Nonsingular plane filling curves of minimum degree over a finite field and their automorphism groups: Supplements to a work of Tallini

Our concern is a nonsingular plane curve defined over a finite field Fq which includes all the Fq-rational points of the projective plane. The possible degree of such a curve is at least q+2. We prove that nonsingular plane curves of degree q+2 having the property actually exist. More precisely, we write down explicitly all of those curves. Actually, Giuseppe Tallini studied such curves in his old paper in 1961. We explain the connection between his work and ours. Also we give another proof of his result on the automorphism group of such a curve, from the viewpoint of linear algebra.