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

Abstract

Our concern is a nonsingular plane curve defined over a finite field F-q which includes all the F-q-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 [8]. 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. (C) 2012 Elsevier Inc. All rights reserved.